let f be PartFunc of REAL,COMPLEX; :: thesis: for A being Subset of REAL holds Im (f | A) = (Im f) | A

let A be Subset of REAL; :: thesis: Im (f | A) = (Im f) | A

.= (dom f) /\ A by COMSEQ_3:def 4

.= dom (f | A) by RELAT_1:61

.= dom (Im (f | A)) by COMSEQ_3:def 4 ;

hence Im (f | A) = (Im f) | A by A1, FUNCT_1:2; :: thesis: verum

let A be Subset of REAL; :: thesis: Im (f | A) = (Im f) | A

A1: now :: thesis: for c being object st c in dom ((Im f) | A) holds

((Im f) | A) . c = (Im (f | A)) . c

dom ((Im f) | A) =
(dom (Im f)) /\ A
by RELAT_1:61
((Im f) | A) . c = (Im (f | A)) . c

let c be object ; :: thesis: ( c in dom ((Im f) | A) implies ((Im f) | A) . c = (Im (f | A)) . c )

assume A2: c in dom ((Im f) | A) ; :: thesis: ((Im f) | A) . c = (Im (f | A)) . c

then A3: c in (dom (Im f)) /\ A by RELAT_1:61;

then A4: c in A by XBOOLE_0:def 4;

A5: c in dom (Im f) by A3, XBOOLE_0:def 4;

then c in dom f by COMSEQ_3:def 4;

then c in (dom f) /\ A by A4, XBOOLE_0:def 4;

then A6: c in dom (f | A) by RELAT_1:61;

then c in dom (Im (f | A)) by COMSEQ_3:def 4;

then (Im (f | A)) . c = Im ((f | A) . c) by COMSEQ_3:def 4

.= Im (f . c) by A6, FUNCT_1:47

.= (Im f) . c by A5, COMSEQ_3:def 4 ;

hence ((Im f) | A) . c = (Im (f | A)) . c by A2, FUNCT_1:47; :: thesis: verum

end;assume A2: c in dom ((Im f) | A) ; :: thesis: ((Im f) | A) . c = (Im (f | A)) . c

then A3: c in (dom (Im f)) /\ A by RELAT_1:61;

then A4: c in A by XBOOLE_0:def 4;

A5: c in dom (Im f) by A3, XBOOLE_0:def 4;

then c in dom f by COMSEQ_3:def 4;

then c in (dom f) /\ A by A4, XBOOLE_0:def 4;

then A6: c in dom (f | A) by RELAT_1:61;

then c in dom (Im (f | A)) by COMSEQ_3:def 4;

then (Im (f | A)) . c = Im ((f | A) . c) by COMSEQ_3:def 4

.= Im (f . c) by A6, FUNCT_1:47

.= (Im f) . c by A5, COMSEQ_3:def 4 ;

hence ((Im f) | A) . c = (Im (f | A)) . c by A2, FUNCT_1:47; :: thesis: verum

.= (dom f) /\ A by COMSEQ_3:def 4

.= dom (f | A) by RELAT_1:61

.= dom (Im (f | A)) by COMSEQ_3:def 4 ;

hence Im (f | A) = (Im f) | A by A1, FUNCT_1:2; :: thesis: verum