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) /\ (dom (f2 - f3)) by VFUNCT_1:def 2
.= (dom f1) /\ ((dom f2) /\ (dom f3)) by VFUNCT_1:def 2
.= ((dom f1) /\ (dom f2)) /\ (dom f3) by XBOOLE_1:16
.= (dom (f1 - f2)) /\ (dom f3) by VFUNCT_1:def 2
.= dom ((f1 - f2) + f3) by VFUNCT_1:def 1 ;

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

let c be Element of M; :: thesis: ( c in dom (f1 - (f2 - f3)) implies (f1 - (f2 - f3)) /. c = ((f1 - f2) + f3) /. c )
assume A2: c in dom (f1 - (f2 - f3)) ; :: thesis: (f1 - (f2 - f3)) /. c = ((f1 - f2) + f3) /. c
then c in (dom f1) /\ (dom (f2 - f3)) by VFUNCT_1:def 2;
then A3: c in dom (f2 - f3) by XBOOLE_0:def 4;
c in (dom (f1 - f2)) /\ (dom f3) by A1, A2, VFUNCT_1:def 1;
then A4: c in dom (f1 - f2) by XBOOLE_0:def 4;
thus (f1 - (f2 - f3)) /. c = (f1 /. c) - ((f2 - f3) /. c) by A2, VFUNCT_1:def 2
.= (f1 /. c) - ((f2 /. c) - (f3 /. c)) by A3, VFUNCT_1:def 2
.= ((f1 /. c) - (f2 /. c)) + (f3 /. c) by RLVECT_1:29
.= ((f1 - f2) /. c) + (f3 /. c) by A4, VFUNCT_1:def 2
.= ((f1 - f2) + f3) /. c by A1, A2, VFUNCT_1:def 1 ; :: thesis: verum

end;

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