let Y be set ; :: thesis: for C being non empty set
for f being PartFunc of C,REAL st f | Y is constant holds
( ( for r being Real holds (r (#) f) | Y is bounded ) & (- f) | Y is bounded & (abs f) | Y is bounded )
let C be non empty set ; :: thesis: for f being PartFunc of C,REAL st f | Y is constant holds
( ( for r being Real holds (r (#) f) | Y is bounded ) & (- f) | Y is bounded & (abs f) | Y is bounded )
let f be PartFunc of C,REAL; :: thesis: ( f | Y is constant implies ( ( for r being Real holds (r (#) f) | Y is bounded ) & (- f) | Y is bounded & (abs f) | Y is bounded ) )
assume A1: f | Y is constant ; :: thesis: ( ( for r being Real holds (r (#) f) | Y is bounded ) & (- f) | Y is bounded & (abs f) | Y is bounded )

hereby :: thesis: ( (- f) | Y is bounded & (abs f) | Y is bounded )

let r be Real; :: thesis: (r (#) f) | Y is bounded
(r (#) f) | Y is constant by A1, Th89;
hence (r (#) f) | Y is bounded ; :: thesis: verum

end;

(- f) | Y is constant by A1, Th90;
hence (- f) | Y is bounded ; :: thesis: (abs f) | Y is bounded
(abs f) | Y is constant by A1, Th91;
hence (abs f) | Y is bounded ; :: thesis: verum