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;

theorem Th31:
  f/g = f(#) (g^)
proof
A1: now
    let c be object;
    assume
A2: c in dom (f/g);
    then c in dom f /\ (dom g \ g"{0}) by Def1;
    then c in dom f /\ dom (g^) by Def2;
    then
A3: c in dom (g^) by XBOOLE_0:def 4;
    thus (f/g).c = f.c * (g.c)" by A2,Def1
      .= f.c * (g^).c by A3,Def2
      .= (f (#) (g^)).c by VALUED_1:5;
  end;
  dom (f/g) = dom f /\ (dom g \ g"{0}) by Def1
    .= dom f /\ dom (g^) by Def2
    .= dom (f(#)(g^)) by VALUED_1:def 4;
  hence thesis by A1,FUNCT_1:2;
end;
