theorem
  f is total implies (-f)/.c = - (f/.c) & (|.f.|).c = |. (f/.c) .|
proof
  assume
A1: f is total;
  then |.f.| is total;
  then
A2: dom |.f.| = dom f & dom (|.f.|) = C by VALUED_1:def 11;
  -f is total by A1;
  then dom (-f) = C;
  hence (-f)/.c = - (f/.c) by Th5;
  thus (|.f.|).c = |. (f.c) .| by VALUED_1:18
    .= |. (f/.c) .| by A2,PARTFUN1:def 6;
end;
