let m, n be non empty Element of NAT ; :: thesis: for f, g being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds
( f + g is_continuous_in x & f - g is_continuous_in x )
let f, g be PartFunc of (REAL m),(REAL n); :: thesis: for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds
( f + g is_continuous_in x & f - g is_continuous_in x )
let x be Element of REAL m; :: thesis: ( f is_continuous_in x & g is_continuous_in x implies ( f + g is_continuous_in x & f - g is_continuous_in x ) )
assume A1: ( f is_continuous_in x & g is_continuous_in x ) ; :: thesis: ( f + g is_continuous_in x & f - g is_continuous_in x )
reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def 4;
A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def 4;
then reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),(REAL-NS n) ;
( f1 is_continuous_in y & g1 is_continuous_in y ) by A1, PDIFF_7:35;
then A2: ( f1 + g1 is_continuous_in y & f1 - g1 is_continuous_in y ) by NFCONT_1:15;
( f + g = f1 + g1 & f - g = f1 - g1 ) by NFCONT_4:5, NFCONT_4:10, A20;
hence ( f + g is_continuous_in x & f - g is_continuous_in x ) by A2, PDIFF_7:35; :: thesis: verum