
theorem
  for f being complex-valued Function
   for c being object holds |.f.|.c = |.f.c.|
proof
  let f be complex-valued Function;
  let c be object;
A1: dom |.f.| = dom f by Def11;
  per cases;
  suppose
    c in dom f;
    hence thesis by A1,Def11;
  end;
  suppose
A2: not c in dom f;
    hence |.f.|.c = |.0 qua Complex.| by A1,COMPLEX1:44,FUNCT_1:def 2
      .= |.f.c.| by A2,FUNCT_1:def 2;
  end;
end;
