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;
