
theorem
for f be PartFunc of REAL,COMPLEX,
    A be Subset of REAL holds Im (f|A) = (Im f)|A
proof
let f be PartFunc of REAL,COMPLEX,
    A be Subset of REAL;
A1: now
    let c be object;
    assume
A2: c in dom ((Im f)|A); 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;
  end;
  dom ((Im f)|A) = dom (Im f) /\ A by RELAT_1:61
    .= 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 thesis by A1,FUNCT_1:2;
end;
