reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th46:
  (-f)|X = -(f|X) & (f^)|X = (f|X)^ & (abs(f))|X = abs((f|X))
proof
A1: now
    let c be object;
    assume
A2: c in dom ((-f)|X);
    then
A3: c in dom (-f) /\ X by RELAT_1:61;
    then c in dom (-f) by XBOOLE_0:def 4;
    then
A4: c in dom f by VALUED_1:8;
    c in X by A3,XBOOLE_0:def 4;
    then c in dom f /\ X by A4,XBOOLE_0:def 4;
    then
A5: c in dom (f|X) by RELAT_1:61;
    thus ((-f)|X).c = (-f).c by A2,FUNCT_1:47
      .= -(f.c) by VALUED_1:8
      .= -((f|X).c) by A5,FUNCT_1:47
      .= (-(f|X)).c by VALUED_1:8;
  end;
  dom ((-f)|X) = dom (-f) /\ X by RELAT_1:61
    .= dom f /\ X by VALUED_1:8
    .= dom (f|X) by RELAT_1:61
    .= dom (-(f|X)) by VALUED_1:8;
  hence (-f)|X = -(f|X) by A1,FUNCT_1:2;
A6: dom ((f^)|X) = dom (f^) /\ X by RELAT_1:61
    .= (dom f \ f"{0}) /\ X by Def2
    .= dom f /\ X \ f"{0} /\ X by XBOOLE_1:50
    .= dom (f|X) \ X /\ f"{0} by RELAT_1:61
    .= dom (f|X) \ (f|X)"{0} by FUNCT_1:70
    .= dom ((f|X)^) by Def2;
A7: dom ((f|X)^) c= dom (f|X) by Th1;
  now
    let c be object;
    assume
A8: c in dom ((f^)|X);
    then c in dom (f^) /\ X by RELAT_1:61;
    then
A9: c in dom (f^) by XBOOLE_0:def 4;
    thus ((f^)|X).c = (f^).c by A8,FUNCT_1:47
      .= (f.c)" by A9,Def2
      .= ((f|X).c)" by A7,A6,A8,FUNCT_1:47
      .= ((f|X)^).c by A6,A8,Def2;
  end;
  hence (f^)|X = (f|X)^ by A6,FUNCT_1:2;
A10: dom ((abs(f))|X) = dom (abs(f)) /\ X by RELAT_1:61
    .= dom f /\ X by VALUED_1:def 11
    .= dom (f|X) by RELAT_1:61
    .= dom (abs((f|X))) by VALUED_1:def 11;
  now
    let c be object;
    assume
A11: c in dom ((abs(f))|X);
    then
A12: c in dom (f|X) by A10,VALUED_1:def 11;
    thus ((abs(f))|X).c = (abs(f)).c by A11,FUNCT_1:47
      .= |.f.c.| by VALUED_1:18
      .= |.(f|X).c.| by A12,FUNCT_1:47
      .= (abs((f|X))).c by VALUED_1:18;
  end;
  hence thesis by A10,FUNCT_1:2;
end;
