let n be non zero Element of NAT ; :: thesis: for a, b being Real
for f being continuous PartFunc of REAL,(REAL-NS n) st dom f = ['a,b'] holds
f | ['a,b'] is bounded
let a, b be Real; :: thesis: for f being continuous PartFunc of REAL,(REAL-NS n) st dom f = ['a,b'] holds
f | ['a,b'] is bounded
let f be continuous PartFunc of REAL,(REAL-NS n); :: thesis: ( dom f = ['a,b'] implies f | ['a,b'] is bounded )
assume A1: dom f = ['a,b'] ; :: thesis: f | ['a,b'] is bounded
reconsider g = f as PartFunc of REAL,(REAL n) by REAL_NS1:def 4;
g is continuous by NFCONT_4:23;
then g | ['a,b'] is bounded by A1, Th29;
hence f | ['a,b'] is bounded by INTEGR19:34; :: thesis: verum