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
  (f/g)(#)g = (f|dom(g^))
proof
A1: dom ((f/g)(#)g) = dom (f/g) /\ dom g by VALUED_1:def 4
    .= dom f /\ (dom g \ g"{0}) /\ dom g by Def1
    .= dom f /\ ((dom g \ g"{0}) /\ dom g) by XBOOLE_1:16
    .= dom f /\ (dom (g^) /\ dom g) by Def2
    .= dom f /\ dom (g^) by Th1,XBOOLE_1:28
    .= dom (f|(dom (g^))) by RELAT_1:61;
  now
    let c be object;
    assume
A2: c in dom ((f/g)(#)g);
    then c in dom f /\ dom (g^) by A1,RELAT_1:61;
    then
A3: c in dom (g^) by XBOOLE_0:def 4;
    then
A4: g.c <> 0 by Th3;
    thus ((f/g)(#)g).c = ((f/g).c) * g.c by VALUED_1:5
      .= (f(#)(g^)).c * g.c by Th31
      .= (f.c) *((g^).c) * g.c by VALUED_1:5
      .= (f.c)*(g.c)"*g.c by A3,Def2
      .= (f.c)*((g.c)" * (g.c))
      .= (f.c)*1 by A4,XCMPLX_0:def 7
      .= (f|(dom (g^))).c by A1,A2,FUNCT_1:47;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
