let M be non empty set ; :: thesis: for V being ComplexNormSpace

for f1, f2 being PartFunc of M,V

for X being set holds

( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,V

for X being set 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 M,V; :: thesis: for X being set holds

( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

let X be set ; :: thesis: ( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

.= ((dom f1) /\ (dom f2)) /\ (X /\ X) by VFUNCT_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_1:61

.= ((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 VFUNCT_1:def 1 ;

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) )

.= ((dom f1) /\ (dom f2)) /\ X by VFUNCT_1:def 1

.= ((dom f1) /\ X) /\ (dom f2) by XBOOLE_1:16

.= (dom (f1 | X)) /\ (dom f2) by RELAT_1:61

.= dom ((f1 | X) + f2) by VFUNCT_1:def 1 ;

hence (f1 + f2) | X = (f1 | X) + f2 by A10, PARTFUN2:1; :: thesis: (f1 + f2) | X = f1 + (f2 | X)

.= ((dom f1) /\ (dom f2)) /\ X by VFUNCT_1:def 1

.= (dom f1) /\ ((dom f2) /\ X) by XBOOLE_1:16

.= (dom f1) /\ (dom (f2 | X)) by RELAT_1:61

.= dom (f1 + (f2 | X)) by VFUNCT_1:def 1 ;

hence (f1 + f2) | X = f1 + (f2 | X) by A18, PARTFUN2:1; :: thesis: verum

for f1, f2 being PartFunc of M,V

for X being set holds

( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,V

for X being set 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 M,V; :: thesis: for X being set holds

( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

let X be set ; :: 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 M st c in dom ((f1 + f2) | X) holds

((f1 + f2) | X) /. c = ((f1 | X) + (f2 | X)) /. c

dom ((f1 + f2) | X) =
(dom (f1 + f2)) /\ X
by RELAT_1:61
((f1 + f2) | X) /. c = ((f1 | X) + (f2 | X)) /. c

let c be Element of M; :: 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 VFUNCT_1:def 1;

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 VFUNCT_1:def 1;

thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A2, PARTFUN2:15

.= (f1 /. c) + (f2 /. c) by A5, VFUNCT_1:def 1

.= ((f1 | X) /. c) + (f2 /. c) by A8, PARTFUN2:15

.= ((f1 | X) /. c) + ((f2 | X) /. c) by A7, PARTFUN2:15

.= ((f1 | X) + (f2 | X)) /. c by A9, VFUNCT_1:def 1 ; :: thesis: verum

end;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 VFUNCT_1:def 1;

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 VFUNCT_1:def 1;

thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A2, PARTFUN2:15

.= (f1 /. c) + (f2 /. c) by A5, VFUNCT_1:def 1

.= ((f1 | X) /. c) + (f2 /. c) by A8, PARTFUN2:15

.= ((f1 | X) /. c) + ((f2 | X) /. c) by A7, PARTFUN2:15

.= ((f1 | X) + (f2 | X)) /. c by A9, VFUNCT_1:def 1 ; :: thesis: verum

.= ((dom f1) /\ (dom f2)) /\ (X /\ X) by VFUNCT_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_1:61

.= ((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 VFUNCT_1:def 1 ;

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 M st c in dom ((f1 + f2) | X) holds

((f1 + f2) | X) /. c = ((f1 | X) + f2) /. c

dom ((f1 + f2) | X) =
(dom (f1 + f2)) /\ X
by RELAT_1:61
((f1 + f2) | X) /. c = ((f1 | X) + f2) /. c

let c be Element of M; :: 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 VFUNCT_1:def 1;

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 VFUNCT_1:def 1;

thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A11, PARTFUN2:15

.= (f1 /. c) + (f2 /. c) by A14, VFUNCT_1:def 1

.= ((f1 | X) /. c) + (f2 /. c) by A16, PARTFUN2:15

.= ((f1 | X) + f2) /. c by A17, VFUNCT_1:def 1 ; :: thesis: verum

end;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 VFUNCT_1:def 1;

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 VFUNCT_1:def 1;

thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A11, PARTFUN2:15

.= (f1 /. c) + (f2 /. c) by A14, VFUNCT_1:def 1

.= ((f1 | X) /. c) + (f2 /. c) by A16, PARTFUN2:15

.= ((f1 | X) + f2) /. c by A17, VFUNCT_1:def 1 ; :: thesis: verum

.= ((dom f1) /\ (dom f2)) /\ X by VFUNCT_1:def 1

.= ((dom f1) /\ X) /\ (dom f2) by XBOOLE_1:16

.= (dom (f1 | X)) /\ (dom f2) by RELAT_1:61

.= dom ((f1 | X) + f2) by VFUNCT_1:def 1 ;

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 M st c in dom ((f1 + f2) | X) holds

((f1 + f2) | X) /. c = (f1 + (f2 | X)) /. c

dom ((f1 + f2) | X) =
(dom (f1 + f2)) /\ X
by RELAT_1:61
((f1 + f2) | X) /. c = (f1 + (f2 | X)) /. c

let c be Element of M; :: 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 VFUNCT_1:def 1;

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 VFUNCT_1:def 1;

thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A19, PARTFUN2:15

.= (f1 /. c) + (f2 /. c) by A22, VFUNCT_1:def 1

.= (f1 /. c) + ((f2 | X) /. c) by A24, PARTFUN2:15

.= (f1 + (f2 | X)) /. c by A25, VFUNCT_1:def 1 ; :: thesis: verum

end;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 VFUNCT_1:def 1;

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 VFUNCT_1:def 1;

thus ((f1 + f2) | X) /. c = (f1 + f2) /. c by A19, PARTFUN2:15

.= (f1 /. c) + (f2 /. c) by A22, VFUNCT_1:def 1

.= (f1 /. c) + ((f2 | X) /. c) by A24, PARTFUN2:15

.= (f1 + (f2 | X)) /. c by A25, VFUNCT_1:def 1 ; :: thesis: verum

.= ((dom f1) /\ (dom f2)) /\ X by VFUNCT_1:def 1

.= (dom f1) /\ ((dom f2) /\ X) by XBOOLE_1:16

.= (dom f1) /\ (dom (f2 | X)) by RELAT_1:61

.= dom (f1 + (f2 | X)) by VFUNCT_1:def 1 ;

hence (f1 + f2) | X = f1 + (f2 | X) by A18, PARTFUN2:1; :: thesis: verum