let X be non empty set ; :: thesis: for f being PartFunc of X,ExtREAL
for c being Real st f is nonnegative holds
( ( 0 <= c implies c (#) f is nonnegative ) & ( c <= 0 implies c (#) f is nonpositive ) )
let f be PartFunc of X,ExtREAL; :: thesis: for c being Real st f is nonnegative holds
( ( 0 <= c implies c (#) f is nonnegative ) & ( c <= 0 implies c (#) f is nonpositive ) )
let c be Real; :: thesis: ( f is nonnegative implies ( ( 0 <= c implies c (#) f is nonnegative ) & ( c <= 0 implies c (#) f is nonpositive ) ) )
set g = c (#) f;
assume A1: f is nonnegative ; :: thesis: ( ( 0 <= c implies c (#) f is nonnegative ) & ( c <= 0 implies c (#) f is nonpositive ) )

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

set g = c (#) f;
assume A2: 0 <= c ; :: thesis: c (#) f is nonnegative
for x being object st x in dom (c (#) f) holds
0 <= (c (#) f) . x

proof

let x be object ; :: thesis: ( x in dom (c (#) f) implies 0 <= (c (#) f) . x )
0 <= f . x by A1, SUPINF_2:51;
then A3: 0 <= c * (f . x) by A2;
assume x in dom (c (#) f) ; :: thesis: 0 <= (c (#) f) . x
hence 0 <= (c (#) f) . x by A3, MESFUNC1:def 6; :: thesis: verum

end;

hence c (#) f is nonnegative by SUPINF_2:52; :: thesis: verum

end;

assume A4: c <= 0 ; :: thesis: c (#) f is nonpositive

now :: thesis: for x being set st x in dom (c (#) f) holds
(c (#) f) . x <= 0

let x be set ; :: thesis: ( x in dom (c (#) f) implies (c (#) f) . x <= 0 )
0 <= f . x by A1, SUPINF_2:51;
then A5: c * (f . x) <= 0 by A4;
assume x in dom (c (#) f) ; :: thesis: (c (#) f) . x <= 0
hence (c (#) f) . x <= 0 by A5, MESFUNC1:def 6; :: thesis: verum

end;

hence c (#) f is nonpositive by Th9; :: thesis: verum