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 is total implies (-f)/.c = - f/.c & (||.f.||).c = ||. f/.c .||
proof
  assume
A1: f is total;
  then dom (-f) = C by PARTFUN1:def 2;
  hence (-f)/.c = - f/.c by Def5;
  ||.f.|| is total by A1,NORMSP_0:def 3;
  hence thesis by NORMSP_0:def 3;
end;
