reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem Th29:
  (-f)|X = -(f|X) & (||.f.||)|X = ||.(f|X).||
proof
A1: now
    let c;
    assume
A2: c in dom ((-f)|X);
    then
A3: c in dom (-f) /\ X by RELAT_1:61;
    then
A4: c in X by XBOOLE_0:def 4;
A5: c in dom (-f) by A3,XBOOLE_0:def 4;
    then c in dom f by Def5;
    then c in dom f /\ X by A4,XBOOLE_0:def 4;
    then
A6: c in dom (f|X) by RELAT_1:61;
    then
A7: c in dom (-(f|X)) by Def5;
    thus ((-f)|X)/.c = (-f)/.c by A2,PARTFUN2:15
      .= -(f/.c) by A5,Def5
      .= -((f|X)/.c) by A6,PARTFUN2:15
      .= (-(f|X))/.c by A7,Def5;
  end;
  dom ((-f)|X) = dom (-f) /\ X by RELAT_1:61
    .= dom f /\ X by Def5
    .= dom (f|X) by RELAT_1:61
    .= dom (-(f|X)) by Def5;
  hence (-f)|X = -(f|X) by A1,PARTFUN2:1;
A8: dom ((||.f.||)|X) = dom (||.f.||) /\ X by RELAT_1:61
    .= dom f /\ X by NORMSP_0:def 3
    .= dom (f|X) by RELAT_1:61
    .= dom (||.(f|X).||) by NORMSP_0:def 3;
  now
    let c;
    assume
A9: c in dom ((||.f.||)|X);
    then
A10: c in dom (f|X) by A8,NORMSP_0:def 3;
    c in dom (||.f.||) /\ X by A9,RELAT_1:61;
    then
A11: c in dom (||.f.||) by XBOOLE_0:def 4;
    thus ((||.f.||)|X).c = (||.f.||).c by A9,FUNCT_1:47
      .= ||.f/.c.|| by A11,NORMSP_0:def 3
      .= ||.(f|X)/.c.|| by A10,PARTFUN2:15
      .= (||.(f|X).||).c by A8,A9,NORMSP_0:def 3;
  end;
  hence thesis by A8,PARTFUN1:5;
end;
