reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;

theorem Th20:
  (Re f)|A = Re(f|A) & (Im f)|A = Im(f|A)
proof
  dom((Re f)|A) = (dom Re f) /\ A by RELAT_1:61
    .= dom f /\ A by COMSEQ_3:def 3;
  then
A1: dom((Re f)|A) = dom(f|A) by RELAT_1:61
    .= dom Re(f|A) by COMSEQ_3:def 3;
  now
    let x be object;
    assume
A2: x in dom(Re(f)|A);
    then
A3: x in dom Re(f) /\ A by RELAT_1:61;
    then
A4: x in dom Re f by XBOOLE_0:def 4;
A5: x in A by A3,XBOOLE_0:def 4;
    thus Re(f|A).x = Re((f|A).x) by A1,A2,COMSEQ_3:def 3
      .= Re(f.x) by A5,FUNCT_1:49
      .= (Re f).x by A4,COMSEQ_3:def 3
      .= ((Re f)|A).x by A5,FUNCT_1:49;
  end;
  hence (Re f)|A = Re(f|A) by A1,FUNCT_1:2;
  dom((Im f)|A) = dom Im f /\ A by RELAT_1:61;
  then dom((Im f)|A) = dom f /\ A by COMSEQ_3:def 4;
  then
A6: dom((Im f)|A) = dom(f|A) by RELAT_1:61
    .= dom Im(f|A) by COMSEQ_3:def 4;
  now
    let x be object;
    assume
A7: x in dom(Im(f)|A);
    then
A8: x in dom Im f /\ A by RELAT_1:61;
    then
A9: x in dom Im f by XBOOLE_0:def 4;
A10: x in A by A8,XBOOLE_0:def 4;
    thus Im(f|A).x = Im((f|A).x) by A6,A7,COMSEQ_3:def 4
      .= Im(f.x) by A10,FUNCT_1:49
      .= (Im f).x by A9,COMSEQ_3:def 4
      .= ((Im f)|A).x by A10,FUNCT_1:49;
  end;
  hence (Im f)|A = Im(f|A) by A6,FUNCT_1:2;
end;
