 reserve Omega for non empty set;
 reserve F for SigmaField of Omega;

theorem
  for RV being random_variable of F,Borel_Sets,
      K being Element of REAL,
      g2 being Function of Omega,REAL
     st g2=chi((RV-(Omega-->K))"[.0,+infty.[,Omega) holds
   Call-Option(RV,K) = g2(#)(RV-(Omega-->K))
proof
 let RV be random_variable of F,Borel_Sets;
 let K be Element of REAL;
 let g2 be Function of Omega,REAL;
 assume A2: g2=chi((RV-(Omega-->K))"[.0,+infty.[,Omega);
 set RVO=RV-(Omega-->K);
 set CO=Call-Option(RV,K);
QQ: dom g2=Omega & dom RVO=Omega & dom CO = Omega by FUNCT_2:def 1;
q1: for c being object st c in dom CO holds CO.c = g2.c * RVO.c
   proof
     let c be object;
     assume c in dom CO;
     then reconsider c as Element of Omega;
      per cases;
      suppose ASSJJ0: c in RVO"[.0,+infty.[; then
       ZW0: g2. c =1 by A2,FUNCT_3:def 3;
       RVO.c in [.0,+infty.[ by ASSJJ0,FUNCT_1:def 7;
       then 0<=RVO.c & RVO. c <+infty by XXREAL_1:3;
       hence thesis by ZW0,FINANCE3:def 5;
      end;
      suppose ASSJJ00: not c in RVO"[.0,+infty.[;
      ASSJJ0: c in RVO"].-infty,0 .[
      proof
        not (c in dom RVO & RVO.c in [.0,+infty.[) by FUNCT_1:def 7,ASSJJ00;
        then not c in Omega or not RVO.c in [.0,+infty.[ by FUNCT_2:def 1;
        then T: RV.c in ].-infty,+infty.[ & -infty<RVO.c & RVO.c <0
          by XXREAL_1:224,XXREAL_0:12,XXREAL_0:9,XXREAL_1:3;
        RVO.c in ].-infty,0 .[ by XXREAL_1:4,T;
        hence thesis by FUNCT_1:def 7,QQ;
      end;
ZW0:  g2. c =0 by A2,ASSJJ00,FUNCT_3:def 3;
      c in dom RVO & RVO.c in ].-infty,0 .[ by ASSJJ0,FUNCT_1:def 7;
      then -infty<RVO.c & RVO. c <0 by XXREAL_1:4;
      hence thesis by ZW0,FINANCE3:def 5;
      end;
   end;
   for x being object st x in dom CO holds CO.x = (g2(#)RVO).x
   proof
     let x be object;
     assume x in dom CO; then
     CO.x=g2.x*RVO.x by q1;
     hence thesis by VALUED_1:5;
   end;
   hence thesis by QQ;
end;
