let n be Element of NAT ; :: thesis: for f1, f2 being PartFunc of REAL, the carrier of (REAL-NS n)
for g1, g2 being PartFunc of REAL,(REAL n) st f1 = g1 & f2 = g2 holds
f1 + f2 = g1 + g2
let f1, f2 be PartFunc of REAL, the carrier of (REAL-NS n); :: thesis: for g1, g2 being PartFunc of REAL,(REAL n) st f1 = g1 & f2 = g2 holds
f1 + f2 = g1 + g2
let g1, g2 be PartFunc of REAL,(REAL n); :: thesis: ( f1 = g1 & f2 = g2 implies f1 + f2 = g1 + g2 )
assume AS: ( f1 = g1 & f2 = g2 ) ; :: thesis: f1 + f2 = g1 + g2
C1: dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 1;
then A2: dom (f1 + f2) = dom (g1 + g2) by AS, VALUED_2:def 45;

B1: now

let x be Element of REAL ; :: thesis: ( x in dom (f1 + f2) implies (f1 + f2) . x = (g1 + g2) . x )
assume A3: x in dom (f1 + f2) ; :: thesis: (f1 + f2) . x = (g1 + g2) . x
then V: ( x in dom g1 & x in dom g2 ) by C1, AS, XBOOLE_0:def 4;
W: ( f1 /. x = g1 /. x & f2 /. x = g2 /. x ) by AS, REAL_NS1:def 4;
( g1 /. x = g1 . x & g2 /. x = g2 . x ) by V, PARTFUN1:def 6;
then R: (f1 /. x) + (f2 /. x) = (g1 . x) + (g2 . x) by W, REAL_NS1:2;
(f1 + f2) /. x = (f1 /. x) + (f2 /. x) by A3, VFUNCT_1:def 1;
then (f1 + f2) . x = (f1 /. x) + (f2 /. x) by A3, PARTFUN1:def 6;
hence (f1 + f2) . x = (g1 + g2) . x by R, A2, A3, VALUED_2:def 45; :: thesis: verum

end;

f1 + f2 is PartFunc of REAL,(REAL n) by REAL_NS1:def 4;
hence f1 + f2 = g1 + g2 by A2, B1, PARTFUN1:5; :: thesis: verum