reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem
  for x be Element of M st f1 is total & f2 is total holds (f1+f2)/.x =
  (f1/.x) + (f2/.x) & (f1-f2)/.x = (f1/.x) - (f2/.x)
proof
  let x be Element of M;
  assume
A1: f1 is total & f2 is total;
  then f1+f2 is total by Th32;
  then dom (f1+f2) = M;
  hence (f1+f2)/.x = (f1/.x) + (f2/.x) by VFUNCT_1:def 1;
  f1-f2 is total by A1,Th32;
  then dom (f1-f2) = M;
  hence thesis by VFUNCT_1:def 2;
end;
