let n be Element of NAT ; :: thesis: for W being non empty set
for f1, f2 being PartFunc of W,(REAL-NS n)
for g1, g2 being PartFunc of W,(REAL n) st f1 = g1 & f2 = g2 holds
f1 - f2 = g1 - g2
let W be non empty set ; :: thesis: for f1, f2 being PartFunc of W,(REAL-NS n)
for g1, g2 being PartFunc of W,(REAL n) st f1 = g1 & f2 = g2 holds
f1 - f2 = g1 - g2
let f1, f2 be PartFunc of W,(REAL-NS n); :: thesis: for g1, g2 being PartFunc of W,(REAL n) st f1 = g1 & f2 = g2 holds
f1 - f2 = g1 - g2
let g1, g2 be PartFunc of W,(REAL n); :: thesis: ( f1 = g1 & f2 = g2 implies f1 - f2 = g1 - g2 )
assume A1: ( f1 = g1 & f2 = g2 ) ; :: thesis: f1 - f2 = g1 - g2
A2: dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 2;
then A3: dom (f1 - f2) = dom (g1 - g2) by A1, VALUED_2:def 46;

A4: now :: thesis: for x being Element of W st x in dom (f1 - f2) holds
(f1 - f2) . x = (g1 - g2) . x

let x be Element of W; :: thesis: ( x in dom (f1 - f2) implies (f1 - f2) . x = (g1 - g2) . x )
assume A5: x in dom (f1 - f2) ; :: thesis: (f1 - f2) . x = (g1 - g2) . x
A6: ( f1 /. x = g1 /. x & f2 /. x = g2 /. x ) by A1, REAL_NS1:def 4;
( x in dom f1 & x in dom f2 ) by A5, A2, XBOOLE_0:def 4;
then ( g1 . x = g1 /. x & g2 /. x = g2 . x ) by A1, PARTFUN1:def 6;
then A7: (f1 /. x) - (f2 /. x) = (g1 . x) - (g2 . x) by A6, REAL_NS1:5;
A8: (f1 - f2) /. x = (f1 /. x) - (f2 /. x) by A5, VFUNCT_1:def 2;
(f1 - f2) /. x = (f1 - f2) . x by A5, PARTFUN1:def 6;
hence (f1 - f2) . x = (g1 - g2) . x by A7, A8, A3, A5, VALUED_2:def 46; :: thesis: verum

end;

f1 - f2 is PartFunc of W,(REAL n) by REAL_NS1:def 4;
hence f1 - f2 = g1 - g2 by A3, A4, PARTFUN1:5; :: thesis: verum