let n, m be Nat; :: thesis: for f1, f2 being PartFunc of (REAL m),(REAL n)
for g1, g2 being PartFunc of (REAL-NS m),(REAL-NS n) st f1 = g1 & f2 = g2 holds
f1 + f2 = g1 + g2
let f1, f2 be PartFunc of (REAL m),(REAL n); :: thesis: for g1, g2 being PartFunc of (REAL-NS m),(REAL-NS n) st f1 = g1 & f2 = g2 holds
f1 + f2 = g1 + g2
let g1, g2 be PartFunc of (REAL-NS m),(REAL-NS n); :: thesis: ( f1 = g1 & f2 = g2 implies f1 + f2 = g1 + g2 )
assume A1: ( f1 = g1 & f2 = g2 ) ; :: thesis: f1 + f2 = g1 + g2
( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def 4;
then reconsider g12 = g1 + g2 as PartFunc of (REAL m),(REAL n) ;
A2: (dom f1) /\ (dom f2) = dom g12 by A1, VFUNCT_1:def 1;
A3: f1 <++> f2 = f1 + f2 by INTEGR15:def 9;
for c being object st c in dom g12 holds
(g1 + g2) . c = (f1 . c) + (f2 . c)

proof

let c be object ; :: thesis: ( c in dom g12 implies (g1 + g2) . c = (f1 . c) + (f2 . c) )
assume A4: c in dom g12 ; :: thesis: (g1 + g2) . c = (f1 . c) + (f2 . c)
then A5: c in dom (g1 + g2) ;
( c in dom f1 & c in dom f2 ) by A2, A4, XBOOLE_0:def 4;
then A6: ( f1 /. c = f1 . c & f2 /. c = f2 . c ) by PARTFUN1:def 6;
A7: ( f1 /. c = g1 /. c & f2 /. c = g2 /. c ) by A1, REAL_NS1:def 4;
g12 . c = (g1 + g2) /. c by A4, PARTFUN1:def 6
.= (g1 /. c) + (g2 /. c) by A5, VFUNCT_1:def 1 ;
hence (g1 + g2) . c = (f1 . c) + (f2 . c) by A6, A7, REAL_NS1:2; :: thesis: verum

end;

hence f1 + f2 = g1 + g2 by A2, A3, VALUED_2:def 45; :: thesis: verum