reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th53:
  (-f)|X = -(f|X) & (f^)|X = (f|X)^ & (|.f.|)|X = |.(f|X).|
proof
A1: dom ((f^)|X) = dom (f^) /\ X by RELAT_1:61
    .= (dom f \ f"{0c}) /\ X by Def2
    .= dom f /\ X \ f"{0c} /\ X by XBOOLE_1:50
    .= dom (f|X) \ X /\ f"{0c} by RELAT_1:61
    .= dom (f|X) \ (f|X)"{0c} by FUNCT_1:70
    .= dom ((f|X)^) by Def2;
A2: now
    let c;
    assume
A3: c in dom ((-f)|X);
    then
A4: c in dom (-f) /\ X by RELAT_1:61;
    then
A5: c in X by XBOOLE_0:def 4;
A6: c in dom (-f) by A4,XBOOLE_0:def 4;
    then c in dom f by Th5;
    then c in dom f /\ X by A5,XBOOLE_0:def 4;
    then
A7: c in dom (f|X) by RELAT_1:61;
    then
A8: c in dom (-(f|X)) by Th5;
    thus ((-f)|X)/.c = (-f)/.c by A3,PARTFUN2:15
      .= -((f/.c)) by A6,Th5
      .= -((f|X)/.c) by A7,PARTFUN2:15
      .= (-(f|X))/.c by A8,Th5;
  end;
  dom ((-f)|X) = dom (-f) /\ X by RELAT_1:61
    .= dom f /\ X by Th5
    .= dom (f|X) by RELAT_1:61
    .= dom (-(f|X)) by Th5;
  hence (-f)|X = -(f|X) by A2,PARTFUN2:1;
A9: dom ((f|X)^) c= dom (f|X) by Th6;
  now
    let c;
    assume
A10: c in dom ((f^)|X);
    then c in dom (f^) /\ X by RELAT_1:61;
    then
A11: c in dom (f^) by XBOOLE_0:def 4;
    thus ((f^)|X)/.c = (f^)/.c by A10,PARTFUN2:15
      .= ((f/.c))" by A11,Def2
      .= ((f|X)/.c)" by A9,A1,A10,PARTFUN2:15
      .= ((f|X)^)/.c by A1,A10,Def2;
  end;
  hence (f^)|X = (f|X)^ by A1,PARTFUN2:1;
A12: dom ((|.f.|)|X) = dom (|.f.|) /\ X by RELAT_1:61
    .= dom f /\ X by VALUED_1:def 11
    .= dom (f|X) by RELAT_1:61
    .= dom (|.(f|X).|) by VALUED_1:def 11;
  now
    let c;
A13: dom |.f.| = dom f by VALUED_1:def 11;
    assume
A14: c in dom ((|.f.|)|X);
    then
A15: c in dom (f|X) by A12,VALUED_1:def 11;
    c in dom (|.f.|) /\ X by A14,RELAT_1:61;
    then
A16: c in dom (|.f.|) by XBOOLE_0:def 4;
    thus ((|.f.|)|X).c = (|.f.|).c by A14,FUNCT_1:47
      .= |.(f.c).| by VALUED_1:18
      .= |.(f/.c).| by A16,A13,PARTFUN1:def 6
      .= |.(f|X)/.c.| by A15,PARTFUN2:15
      .= |.(f|X).c.| by A15,PARTFUN1:def 6
      .= (|.(f|X).|).c by VALUED_1:18;
  end;
  hence thesis by A12,PARTFUN1:5;
end;
