let X be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,the carrier of V holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
let C be non empty set ; :: thesis: for V being RealNormSpace
for f1, f2 being PartFunc of C,the carrier of V holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
let V be RealNormSpace; :: thesis: for f1, f2 being PartFunc of C,the carrier of V holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
let f1, f2 be PartFunc of C,the carrier of V; :: thesis: ( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
A1: dom ((f1 + f2) | X) = (dom (f1 + f2)) /\ X by RELAT_1:90
.= ((dom f1) /\ (dom f2)) /\ (X /\ X) by Def1
.= (dom f1) /\ ((dom f2) /\ (X /\ X)) by XBOOLE_1:16
.= (dom f1) /\ (((dom f2) /\ X) /\ X) by XBOOLE_1:16
.= (dom f1) /\ (X /\ (dom (f2 | X))) by RELAT_1:90
.= ((dom f1) /\ X) /\ (dom (f2 | X)) by XBOOLE_1:16
.= (dom (f1 | X)) /\ (dom (f2 | X)) by RELAT_1:90
.= dom ((f1 | X) + (f2 | X)) by Def1 ;

now

let c be Element of C; :: thesis: ( c in dom ((f1 + f2) | X) implies ((f1 + f2) | X) /. c = ((f1 | X) + (f2 | X)) /. c )
assume A2: c in dom ((f1 + f2) | X) ; :: thesis: ((f1 + f2) | X) /. c = ((f1 | X) + (f2 | X)) /. c
then c in (dom (f1 + f2)) /\ X by RELAT_1:90;
then A3: ( c in dom (f1 + f2) & c in X ) by XBOOLE_0:def 4;
then c in (dom f1) /\ (dom f2) by Def1;
then ( c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then ( c in (dom f1) /\ X & c in (dom f2) /\ X ) by A3, XBOOLE_0:def 4;
then A4: ( c in dom (f1 | X) & c in dom (f2 | X) ) by RELAT_1:90;
then c in (dom (f1 | X)) /\ (dom (f2 | X)) by XBOOLE_0:def 4;
then A5: c in dom ((f1 | X) + (f2 | X)) by Def1;
thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A2, PARTFUN2:32
.= (f1 /. c) + (f2 /. c) by A3, Def1
.= ((f1 | X) /. c) + (f2 /. c) by A4, PARTFUN2:32
.= ((f1 | X) /. c) + ((f2 | X) /. c) by A4, PARTFUN2:32
.= ((f1 | X) + (f2 | X)) /. c by A5, Def1 ; :: thesis: verum

end;

hence (f1 + f2) | X = (f1 | X) + (f2 | X) by A1, PARTFUN2:3; :: thesis: ( (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
A6: dom ((f1 + f2) | X) = (dom (f1 + f2)) /\ X by RELAT_1:90
.= ((dom f1) /\ (dom f2)) /\ X by Def1
.= ((dom f1) /\ X) /\ (dom f2) by XBOOLE_1:16
.= (dom (f1 | X)) /\ (dom f2) by RELAT_1:90
.= dom ((f1 | X) + f2) by Def1 ;

now

let c be Element of C; :: thesis: ( c in dom ((f1 + f2) | X) implies ((f1 + f2) | X) /. c = ((f1 | X) + f2) /. c )
assume A7: c in dom ((f1 + f2) | X) ; :: thesis: ((f1 + f2) | X) /. c = ((f1 | X) + f2) /. c
then c in (dom (f1 + f2)) /\ X by RELAT_1:90;
then A8: ( c in dom (f1 + f2) & c in X ) by XBOOLE_0:def 4;
then c in (dom f1) /\ (dom f2) by Def1;
then ( c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then ( c in (dom f1) /\ X & c in dom f2 ) by A8, XBOOLE_0:def 4;
then A9: ( c in dom (f1 | X) & c in dom f2 ) by RELAT_1:90;
then c in (dom (f1 | X)) /\ (dom f2) by XBOOLE_0:def 4;
then A10: c in dom ((f1 | X) + f2) by Def1;
thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A7, PARTFUN2:32
.= (f1 /. c) + (f2 /. c) by A8, Def1
.= ((f1 | X) /. c) + (f2 /. c) by A9, PARTFUN2:32
.= ((f1 | X) + f2) /. c by A10, Def1 ; :: thesis: verum

end;

hence (f1 + f2) | X = (f1 | X) + f2 by A6, PARTFUN2:3; :: thesis: (f1 + f2) | X = f1 + (f2 | X)
A11: dom ((f1 + f2) | X) = (dom (f1 + f2)) /\ X by RELAT_1:90
.= ((dom f1) /\ (dom f2)) /\ X by Def1
.= (dom f1) /\ ((dom f2) /\ X) by XBOOLE_1:16
.= (dom f1) /\ (dom (f2 | X)) by RELAT_1:90
.= dom (f1 + (f2 | X)) by Def1 ;

now

let c be Element of C; :: thesis: ( c in dom ((f1 + f2) | X) implies ((f1 + f2) | X) /. c = (f1 + (f2 | X)) /. c )
assume A12: c in dom ((f1 + f2) | X) ; :: thesis: ((f1 + f2) | X) /. c = (f1 + (f2 | X)) /. c
then c in (dom (f1 + f2)) /\ X by RELAT_1:90;
then A13: ( c in dom (f1 + f2) & c in X ) by XBOOLE_0:def 4;
then c in (dom f1) /\ (dom f2) by Def1;
then ( c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then ( c in dom f1 & c in (dom f2) /\ X ) by A13, XBOOLE_0:def 4;
then A14: ( c in dom f1 & c in dom (f2 | X) ) by RELAT_1:90;
then c in (dom f1) /\ (dom (f2 | X)) by XBOOLE_0:def 4;
then A15: c in dom (f1 + (f2 | X)) by Def1;
thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A12, PARTFUN2:32
.= (f1 /. c) + (f2 /. c) by A13, Def1
.= (f1 /. c) + ((f2 | X) /. c) by A14, PARTFUN2:32
.= (f1 + (f2 | X)) /. c by A15, Def1 ; :: thesis: verum

end;

hence (f1 + f2) | X = f1 + (f2 | X) by A11, PARTFUN2:3; :: thesis: verum