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 4
.= (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 4 ;

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 4;
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;
thus ((f1 (#) f2) | X) . c = (f1 (#) f2) . c by A2, FUNCT_1:70
.= (f1 . c) * (f2 . c) by VALUED_1:5
.= ((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 VALUED_1:5 ; :: 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) )
A5: dom ((f1 (#) f2) | X) = (dom (f1 (#) f2)) /\ X by RELAT_1:90
.= ((dom f1) /\ (dom f2)) /\ X by VALUED_1:def 4
.= ((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 4 ;

now

let c be set ; :: thesis: ( c in dom ((f1 (#) f2) | X) implies ((f1 (#) f2) | X) . c = ((f1 | X) (#) f2) . c )
assume A6: 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 A7: ( c in dom (f1 (#) f2) & c in X ) by XBOOLE_0:def 4;
then c in (dom f1) /\ (dom f2) by VALUED_1:def 4;
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 A7, XBOOLE_0:def 4;
then A8: ( c in dom (f1 | X) & c in dom f2 ) by RELAT_1:90;
thus ((f1 (#) f2) | X) . c = (f1 (#) f2) . c by A6, FUNCT_1:70
.= (f1 . c) * (f2 . c) by VALUED_1:5
.= ((f1 | X) . c) * (f2 . c) by A8, FUNCT_1:70
.= ((f1 | X) (#) f2) . c by VALUED_1:5 ; :: thesis: verum

end;

hence (f1 (#) f2) | X = (f1 | X) (#) f2 by A5, FUNCT_1:9; :: thesis: (f1 (#) f2) | X = f1 (#) (f2 | X)
A9: dom ((f1 (#) f2) | X) = (dom (f1 (#) f2)) /\ X by RELAT_1:90
.= ((dom f1) /\ (dom f2)) /\ X by VALUED_1:def 4
.= (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 4 ;

now

let c be set ; :: thesis: ( c in dom ((f1 (#) f2) | X) implies ((f1 (#) f2) | X) . c = (f1 (#) (f2 | X)) . c )
assume A10: 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 A11: ( c in dom (f1 (#) f2) & c in X ) by XBOOLE_0:def 4;
then c in (dom f1) /\ (dom f2) by VALUED_1:def 4;
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 A11, XBOOLE_0:def 4;
then A12: ( c in dom f1 & c in dom (f2 | X) ) by RELAT_1:90;
thus ((f1 (#) f2) | X) . c = (f1 (#) f2) . c by A10, FUNCT_1:70
.= (f1 . c) * (f2 . c) by VALUED_1:5
.= (f1 . c) * ((f2 | X) . c) by A12, FUNCT_1:70
.= (f1 (#) (f2 | X)) . c by VALUED_1:5 ; :: thesis: verum

end;

hence (f1 (#) f2) | X = f1 (#) (f2 | X) by A9, FUNCT_1:9; :: thesis: verum