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
  (||.f.||)"{0} = f"{0.V} & (-f)"{0.V} = f"{0.V}
proof
  now
    let c;
    thus c in (||.f.||)"{0} implies c in f"{0.V}
    proof
      assume
A1:   c in (||.f.||)"{0};
      then
A2:   c in dom (||.f.||) by FUNCT_1:def 7;
      (||.f.||).c in {0} by A1,FUNCT_1:def 7;
      then (||.f.||).c = 0 by TARSKI:def 1;
      then ||.f/.c.|| = 0 by A2,NORMSP_0:def 3;
      then f/.c = 0.V by NORMSP_0:def 5;
      then
A3:   f/.c in {0.V} by TARSKI:def 1;
      c in dom f by A2,NORMSP_0:def 3;
      hence thesis by A3,PARTFUN2:26;
    end;
    assume
A4: c in (f)"{0.V};
    then c in dom f by PARTFUN2:26;
    then
A5: c in dom (||.f.||) by NORMSP_0:def 3;
    f/.c in {0.V} by A4,PARTFUN2:26;
    then f/.c = 0.V by TARSKI:def 1;
    then ||.f/.c.|| = 0;
    then (||.f.||).c = 0 by A5,NORMSP_0:def 3;
    then (||.f.||).c in {0} by TARSKI:def 1;
    hence c in (||.f.||)"{0} by A5,FUNCT_1:def 7;
  end;
  hence (||.f.||)"{0} = f"{0.V} by SUBSET_1:3;
  now
    let c;
    thus c in (-f)"{0.V} implies c in f"{0.V}
    proof
      assume
A6:   c in (-f)"{0.V};
      then
A7:   c in dom (-f) by PARTFUN2:26;
      (-f)/.c in {0.V} by A6,PARTFUN2:26;
      then (-f)/.c = 0.V by TARSKI:def 1;
      then --(f/.c) = -0.V by A7,Def5;
      then f/.c = 0.V;
      then
A8:   f/.c in {0.V} by TARSKI:def 1;
      c in dom f by A7,Def5;
      hence thesis by A8,PARTFUN2:26;
    end;
    assume
A9: c in (f)"{0.V};
    then c in dom f by PARTFUN2:26;
    then
A10: c in dom (-f) by Def5;
    f/.c in {0.V} by A9,PARTFUN2:26;
    then f/.c = 0.V by TARSKI:def 1;
    then (-f)/.c = -0.V by A10,Def5;
    then (-f)/.c in {0.V} by TARSKI:def 1;
    hence c in (-f)"{0.V} by A10,PARTFUN2:26;
  end;
  hence thesis by SUBSET_1:3;
end;
