let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,V holds (f1 + f2) + f3 = f1 + (f2 + f3)
let V be ComplexNormSpace; :: thesis: for f1, f2, f3 being PartFunc of M,V holds (f1 + f2) + f3 = f1 + (f2 + f3)
let f1, f2, f3 be PartFunc of M,V; :: thesis: (f1 + f2) + f3 = f1 + (f2 + f3)
A1: dom ((f1 + f2) + f3) = (dom (f1 + f2)) /\ (dom f3) by VFUNCT_1:def 1
.= ((dom f1) /\ (dom f2)) /\ (dom f3) by VFUNCT_1:def 1
.= (dom f1) /\ ((dom f2) /\ (dom f3)) by XBOOLE_1:16
.= (dom f1) /\ (dom (f2 + f3)) by VFUNCT_1:def 1
.= dom (f1 + (f2 + f3)) by VFUNCT_1:def 1 ;

now :: thesis: for x being Element of M st x in dom ((f1 + f2) + f3) holds
((f1 + f2) + f3) /. x = (f1 + (f2 + f3)) /. x

let x be Element of M; :: thesis: ( x in dom ((f1 + f2) + f3) implies ((f1 + f2) + f3) /. x = (f1 + (f2 + f3)) /. x )
assume A2: x in dom ((f1 + f2) + f3) ; :: thesis: ((f1 + f2) + f3) /. x = (f1 + (f2 + f3)) /. x
then x in (dom (f1 + f2)) /\ (dom f3) by VFUNCT_1:def 1;
then A3: x in dom (f1 + f2) by XBOOLE_0:def 4;
x in (dom f1) /\ (dom (f2 + f3)) by A1, A2, VFUNCT_1:def 1;
then A4: x in dom (f2 + f3) by XBOOLE_0:def 4;
thus ((f1 + f2) + f3) /. x = ((f1 + f2) /. x) + (f3 /. x) by A2, VFUNCT_1:def 1
.= ((f1 /. x) + (f2 /. x)) + (f3 /. x) by A3, VFUNCT_1:def 1
.= (f1 /. x) + ((f2 /. x) + (f3 /. x)) by RLVECT_1:def 3
.= (f1 /. x) + ((f2 + f3) /. x) by A4, VFUNCT_1:def 1
.= (f1 + (f2 + f3)) /. x by A1, A2, VFUNCT_1:def 1 ; :: thesis: verum

end;

hence (f1 + f2) + f3 = f1 + (f2 + f3) by A1, PARTFUN2:1; :: thesis: verum