let X be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL 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 f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL 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 f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL holds
( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) & (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )
let f2 be PartFunc of C,V; :: thesis: for f1 being PartFunc of C,REAL holds
( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) & (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )
let f1 be PartFunc of C,REAL; :: thesis: ( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) & (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )

A1: now :: thesis: for c being Element of C st c in dom ((f1 (#) f2) | X) holds
((f1 (#) f2) | X) /. c = ((f1 | X) (#) (f2 | X)) /. c

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 A3: c in (dom (f1 (#) f2)) /\ X by RELAT_1:61;
then A4: c in X by XBOOLE_0:def 4;
A5: c in dom (f1 (#) f2) by A3, XBOOLE_0:def 4;
then A6: c in (dom f1) /\ (dom f2) by Def3;
then c in dom f2 by XBOOLE_0:def 4;
then c in (dom f2) /\ X by A4, XBOOLE_0:def 4;
then A7: c in dom (f2 | X) by RELAT_1:61;
c in dom f1 by A6, XBOOLE_0:def 4;
then c in (dom f1) /\ X by A4, XBOOLE_0:def 4;
then A8: c in dom (f1 | X) by RELAT_1:61;
then c in (dom (f1 | X)) /\ (dom (f2 | X)) by A7, XBOOLE_0:def 4;
then A9: c in dom ((f1 | X) (#) (f2 | X)) by Def3;
thus ((f1 (#) f2) | X) /. c = (f1 (#) f2) /. c by A2, PARTFUN2:15
.= (f1 . c) * (f2 /. c) by A5, Def3
.= ((f1 | X) . c) * (f2 /. c) by A8, FUNCT_1:47
.= ((f1 | X) . c) * ((f2 | X) /. c) by A7, PARTFUN2:15
.= ((f1 | X) (#) (f2 | X)) /. c by A9, Def3 ; :: thesis: verum

end;

dom ((f1 (#) f2) | X) = (dom (f1 (#) f2)) /\ X by RELAT_1:61
.= ((dom f1) /\ (dom f2)) /\ (X /\ X) by Def3
.= (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_1:61
.= ((dom f1) /\ X) /\ (dom (f2 | X)) by XBOOLE_1:16
.= (dom (f1 | X)) /\ (dom (f2 | X)) by RELAT_1:61
.= dom ((f1 | X) (#) (f2 | X)) by Def3 ;
hence (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) by A1, PARTFUN2:1; :: thesis: ( (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )

A10: now :: thesis: for c being Element of C st c in dom ((f1 (#) f2) | X) holds
((f1 (#) f2) | X) /. c = ((f1 | X) (#) f2) /. c

let c be Element of C; :: thesis: ( c in dom ((f1 (#) f2) | X) implies ((f1 (#) f2) | X) /. c = ((f1 | X) (#) f2) /. c )
assume A11: c in dom ((f1 (#) f2) | X) ; :: thesis: ((f1 (#) f2) | X) /. c = ((f1 | X) (#) f2) /. c
then A12: c in (dom (f1 (#) f2)) /\ X by RELAT_1:61;
then A13: c in X by XBOOLE_0:def 4;
A14: c in dom (f1 (#) f2) by A12, XBOOLE_0:def 4;
then A15: c in (dom f1) /\ (dom f2) by Def3;
then c in dom f1 by XBOOLE_0:def 4;
then c in (dom f1) /\ X by A13, XBOOLE_0:def 4;
then A16: c in dom (f1 | X) by RELAT_1:61;
c in dom f2 by A15, XBOOLE_0:def 4;
then c in (dom (f1 | X)) /\ (dom f2) by A16, XBOOLE_0:def 4;
then A17: c in dom ((f1 | X) (#) f2) by Def3;
thus ((f1 (#) f2) | X) /. c = (f1 (#) f2) /. c by A11, PARTFUN2:15
.= (f1 . c) * (f2 /. c) by A14, Def3
.= ((f1 | X) . c) * (f2 /. c) by A16, FUNCT_1:47
.= ((f1 | X) (#) f2) /. c by A17, Def3 ; :: thesis: verum

end;

dom ((f1 (#) f2) | X) = (dom (f1 (#) f2)) /\ X by RELAT_1:61
.= ((dom f1) /\ (dom f2)) /\ X by Def3
.= ((dom f1) /\ X) /\ (dom f2) by XBOOLE_1:16
.= (dom (f1 | X)) /\ (dom f2) by RELAT_1:61
.= dom ((f1 | X) (#) f2) by Def3 ;
hence (f1 (#) f2) | X = (f1 | X) (#) f2 by A10, PARTFUN2:1; :: thesis: (f1 (#) f2) | X = f1 (#) (f2 | X)

A18: now :: thesis: for c being Element of C st c in dom ((f1 (#) f2) | X) holds
((f1 (#) f2) | X) /. c = (f1 (#) (f2 | X)) /. c

let c be Element of C; :: thesis: ( c in dom ((f1 (#) f2) | X) implies ((f1 (#) f2) | X) /. c = (f1 (#) (f2 | X)) /. c )
assume A19: c in dom ((f1 (#) f2) | X) ; :: thesis: ((f1 (#) f2) | X) /. c = (f1 (#) (f2 | X)) /. c
then A20: c in (dom (f1 (#) f2)) /\ X by RELAT_1:61;
then A21: c in X by XBOOLE_0:def 4;
A22: c in dom (f1 (#) f2) by A20, XBOOLE_0:def 4;
then A23: c in (dom f1) /\ (dom f2) by Def3;
then c in dom f2 by XBOOLE_0:def 4;
then c in (dom f2) /\ X by A21, XBOOLE_0:def 4;
then A24: c in dom (f2 | X) by RELAT_1:61;
c in dom f1 by A23, XBOOLE_0:def 4;
then c in (dom f1) /\ (dom (f2 | X)) by A24, XBOOLE_0:def 4;
then A25: c in dom (f1 (#) (f2 | X)) by Def3;
thus ((f1 (#) f2) | X) /. c = (f1 (#) f2) /. c by A19, PARTFUN2:15
.= (f1 . c) * (f2 /. c) by A22, Def3
.= (f1 . c) * ((f2 | X) /. c) by A24, PARTFUN2:15
.= (f1 (#) (f2 | X)) /. c by A25, Def3 ; :: thesis: verum

end;

dom ((f1 (#) f2) | X) = (dom (f1 (#) f2)) /\ X by RELAT_1:61
.= ((dom f1) /\ (dom f2)) /\ X by Def3
.= (dom f1) /\ ((dom f2) /\ X) by XBOOLE_1:16
.= (dom f1) /\ (dom (f2 | X)) by RELAT_1:61
.= dom (f1 (#) (f2 | X)) by Def3 ;
hence (f1 (#) f2) | X = f1 (#) (f2 | X) by A18, PARTFUN2:1; :: thesis: verum