reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th38:
  f/g = f (#) (g^)
proof
A1: now
    let c;
    assume
A2: c in dom (f/g);
    then c in dom f /\ (dom g \ g"{0c}) by Def1;
    then
A3: c in dom f /\ dom (g^) by Def2;
    then
A4: c in dom (g^) by XBOOLE_0:def 4;
A5: c in dom (f (#) (g^)) by A3,Th3;
    thus (f/g)/.c = (f/.c) * (g/.c)" by A2,Def1
      .= (f/.c) * ((g^)/.c) by A4,Def2
      .= (f (#) (g^))/.c by A5,Th3;
  end;
  dom (f/g) = dom f /\ (dom g \ g"{0c}) by Def1
    .= dom f /\ dom (g^) by Def2
    .= dom (f(#)(g^)) by Th3;
  hence thesis by A1,PARTFUN2:1;
end;
