let X be set ; :: thesis: for f1, f2 being complex-valued Function holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
let f1, f2 be complex-valued Function; :: 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 VALUED_1:def 1
.= (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 VALUED_1:def 1 ;

now

let c be set ; :: 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 VALUED_1:def 1;
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 VALUED_1:def 1;
thus ((f1 + f2) | X) . c = (f1 + f2) . c by A2, FUNCT_1:70
.= (f1 . c) + (f2 . c) by A3, VALUED_1:def 1
.= ((f1 | X) . c) + (f2 . c) by A4, FUNCT_1:70
.= ((f1 | X) . c) + ((f2 | X) . c) by A4, FUNCT_1:70
.= ((f1 | X) + (f2 | X)) . c by A5, VALUED_1:def 1 ; :: thesis: verum

end;

hence (f1 + f2) | X = (f1 | X) + (f2 | X) by A1, FUNCT_1:9; :: 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 VALUED_1:def 1
.= ((dom f1) /\ X) /\ (dom f2) by XBOOLE_1:16
.= (dom (f1 | X)) /\ (dom f2) by RELAT_1:90
.= dom ((f1 | X) + f2) by VALUED_1:def 1 ;

now

let c be set ; :: 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 VALUED_1:def 1;
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 VALUED_1:def 1;
thus ((f1 + f2) | X) . c = (f1 + f2) . c by A7, FUNCT_1:70
.= (f1 . c) + (f2 . c) by A8, VALUED_1:def 1
.= ((f1 | X) . c) + (f2 . c) by A9, FUNCT_1:70
.= ((f1 | X) + f2) . c by A10, VALUED_1:def 1 ; :: thesis: verum

end;

hence (f1 + f2) | X = (f1 | X) + f2 by A6, FUNCT_1:9; :: 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 VALUED_1:def 1
.= (dom f1) /\ ((dom f2) /\ X) by XBOOLE_1:16
.= (dom f1) /\ (dom (f2 | X)) by RELAT_1:90
.= dom (f1 + (f2 | X)) by VALUED_1:def 1 ;

now

let c be set ; :: 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 VALUED_1:def 1;
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 VALUED_1:def 1;
thus ((f1 + f2) | X) . c = (f1 + f2) . c by A12, FUNCT_1:70
.= (f1 . c) + (f2 . c) by A13, VALUED_1:def 1
.= (f1 . c) + ((f2 | X) . c) by A14, FUNCT_1:70
.= (f1 + (f2 | X)) . c by A15, VALUED_1:def 1 ; :: thesis: verum

end;

hence (f1 + f2) | X = f1 + (f2 | X) by A11, FUNCT_1:9; :: thesis: verum