let n be Element of NAT ; :: thesis: for X being set
for f being PartFunc of REAL,(REAL n) st X c= dom f & f | X is continuous holds
( |.f.| | X is continuous & (- f) | X is continuous )
let X be set ; :: thesis: for f being PartFunc of REAL,(REAL n) st X c= dom f & f | X is continuous holds
( |.f.| | X is continuous & (- f) | X is continuous )
let f be PartFunc of REAL,(REAL n); :: thesis: ( X c= dom f & f | X is continuous implies ( |.f.| | X is continuous & (- f) | X is continuous ) )
assume AS: ( X c= dom f & f | X is continuous ) ; :: thesis: ( |.f.| | X is continuous & (- f) | X is continuous )
reconsider g = f as PartFunc of REAL, the carrier of (REAL-NS n) by REAL_NS1:def 4;
g | X is continuous by AS, Def2Th;
then P1: ( ||.g.|| | X is continuous & (- g) | X is continuous ) by AS, NFCONT_3:22;
hence |.f.| | X is continuous by LM003F; :: thesis: (- f) | X is continuous
- g = - f by LM003E;
hence (- f) | X is continuous by P1, Def2Th; :: thesis: verum