let X be non empty set ; :: thesis: for f being PartFunc of X,ExtREAL

for c being Real st f is nonpositive holds

( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) )

let f be PartFunc of X,ExtREAL; :: thesis: for c being Real st f is nonpositive holds

( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) )

let c be Real; :: thesis: ( f is nonpositive implies ( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) ) )

set g = c (#) f;

assume A1: f is nonpositive ; :: thesis: ( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) )

for c being Real st f is nonpositive holds

( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) )

let f be PartFunc of X,ExtREAL; :: thesis: for c being Real st f is nonpositive holds

( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) )

let c be Real; :: thesis: ( f is nonpositive implies ( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) ) )

set g = c (#) f;

assume A1: f is nonpositive ; :: thesis: ( ( 0 <= c implies c (#) f is nonpositive ) & ( c <= 0 implies c (#) f is nonnegative ) )

hereby :: thesis: ( c <= 0 implies c (#) f is nonnegative )

assume A4:
c <= 0
; :: thesis: c (#) f is nonnegative set g = c (#) f;

assume A2: 0 <= c ; :: thesis: c (#) f is nonpositive

for x being set st x in dom (c (#) f) holds

(c (#) f) . x <= 0

end;assume A2: 0 <= c ; :: thesis: c (#) f is nonpositive

for x being set st x in dom (c (#) f) holds

(c (#) f) . x <= 0

proof

hence
c (#) f is nonpositive
by MESFUNC5:9; :: thesis: verum
let x be set ; :: thesis: ( x in dom (c (#) f) implies (c (#) f) . x <= 0 )

f . x <= 0 by A1, MESFUNC5:8;

then A3: c * (f . x) <= 0 by A2;

assume x in dom (c (#) f) ; :: thesis: (c (#) f) . x <= 0

hence (c (#) f) . x <= 0 by A3, MESFUNC1:def 6; :: thesis: verum

end;f . x <= 0 by A1, MESFUNC5:8;

then A3: c * (f . x) <= 0 by A2;

assume x in dom (c (#) f) ; :: thesis: (c (#) f) . x <= 0

hence (c (#) f) . x <= 0 by A3, MESFUNC1:def 6; :: thesis: verum

now :: thesis: for x being object st x in dom (c (#) f) holds

0 <= (c (#) f) . x

hence
c (#) f is nonnegative
by SUPINF_2:52; :: thesis: verum0 <= (c (#) f) . x

let x be object ; :: thesis: ( x in dom (c (#) f) implies 0 <= (c (#) f) . x )

f . x <= 0 by A1, MESFUNC5:8;

then A5: 0 <= c * (f . x) by A4;

assume x in dom (c (#) f) ; :: thesis: 0 <= (c (#) f) . x

hence 0 <= (c (#) f) . x by A5, MESFUNC1:def 6; :: thesis: verum

end;f . x <= 0 by A1, MESFUNC5:8;

then A5: 0 <= c * (f . x) by A4;

assume x in dom (c (#) f) ; :: thesis: 0 <= (c (#) f) . x

hence 0 <= (c (#) f) . x by A5, MESFUNC1:def 6; :: thesis: verum