reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem
  f is total implies (r(#)f).c = r * (f.c)
proof
  assume f is total;
  then r(#)f is total;
  then dom (r(#)f) = C;
  hence thesis by VALUED_1:def 5;
end;
