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

theorem
  for RV be random_variable of F,Borel_Sets,
      K be Real
  for w being Element of Omega holds
    Straddle(RV,K).w=|.(RV-(Omega-->K)).w.|
proof
  let RV be random_variable of F,Borel_Sets;
  let K be Real;
  let w be Element of Omega;
  set f1=RV;
  set f2=(-1)(#)(Omega-->K);
SS: dom RV=Omega & dom f2=Omega by FUNCT_2:def 1; then
   w in dom f1/\dom f2; then
C1: w in dom (f1+f2) by VALUED_1:def 1;
  dom Put-Option(RV,K) = Omega &
      dom Call-Option(RV,K) = Omega by FUNCT_2:def 1; then
  w in dom Put-Option(RV,K) /\ dom Call-Option(RV,K); then
 B100: w in dom (Put-Option(RV,K)+Call-Option(RV,K)) by VALUED_1:def 1;
c2: dom(Omega-->K)=Omega & dom f2 = dom(Omega-->K)
     by VALUED_1:def 5;
c4: w in dom(RV) by SS; then
C6: w in dom ((-1)(#)RV) by VALUED_1:def 5;
 per cases;
 suppose S1: (RV-(Omega-->K)).w>0; then
B0: |.(RV-(Omega-->K)).w.|=(RV-(Omega-->K)).w by COMPLEX1:43;
   dom(-RV)=Omega by FUNCT_2:def 1;
   then w in dom(Omega-->K)/\dom(-RV); then
  C7: w in dom((Omega-->K)+ -RV) by VALUED_1:def 1;
 ((Omega-->K)-RV).w<0
 proof
  (RV-(Omega-->K)).w=f1.w + f2.w by C1,VALUED_1:def 1
  .=RV.w + ((-1)*((Omega-->K).w)) by c2,VALUED_1:def 5;
  then -(RV.w-K)<0 by S1;
  then K-RV.w<0;
  then (Omega-->K).w + (-1)*RV.w<0;
  then (Omega-->K).w + ((-1)(#)RV).w<0 by C6,VALUED_1:def 5;
  hence thesis by C7,VALUED_1:def 1;
 end; then
 B2: Put-Option(RV,K).w=0 by Def30;
 (Put-Option(RV,K)+Call-Option(RV,K)).w=Put-Option(RV,K).w+Call-Option(RV,K).w
  by B100,VALUED_1:def 1;
 hence thesis by B0,B2,FINANCE3:def 5,S1;
 end;
 suppose S1: (RV-(Omega-->K)).w<0;
  dom Put-Option(RV,K) = Omega &
    dom Call-Option(RV,K) = Omega by FUNCT_2:def 1; then
  w in dom Put-Option(RV,K) /\ dom Call-Option(RV,K); then
 B100: w in dom (Put-Option(RV,K)+Call-Option(RV,K)) by VALUED_1:def 1;
c2: dom(Omega-->K)=Omega & dom f2 = dom(Omega-->K)
     by VALUED_1:def 5;
C6: w in dom ((-1)(#)RV) by VALUED_1:def 5,c4;
   dom(Omega-->K)=Omega & dom(-RV)=Omega by FUNCT_2:def 1;
   then w in dom(Omega-->K)/\dom(-RV); then
  C7: w in dom((Omega-->K)+ -RV) by VALUED_1:def 1;
 ((Omega-->K)-RV).w>0
 proof
  (RV-(Omega-->K)).w=f1.w + f2.w by C1,VALUED_1:def 1
  .=RV.w + ((-1)*((Omega-->K).w)) by c2,VALUED_1:def 5;
  then -(RV.w-K)>0 by S1;
  then K-RV.w>0;
  then (Omega-->K).w + (-1)*RV.w>0;
  then (Omega-->K).w + ((-1)(#)RV).w>0 by C6,VALUED_1:def 5;
 hence thesis by C7,VALUED_1:def 1;
 end; then
 B2: Put-Option(RV,K).w=((Omega-->K)-RV).w by Def30;
 Call-Option(RV,K).w = 0 by FINANCE3:def 5,S1;
 then B40: Straddle(RV,K).w=0+((Omega-->K)-RV).w by B2,B100,VALUED_1:def 1;
   dom(Omega-->K)=Omega & dom((-1)(#)RV)=Omega by FUNCT_2:def 1;
   then w in dom(Omega-->K) /\ dom((-1)(#)RV); then
  w in dom ((Omega-->K)+ ((-1)(#)RV)) by VALUED_1:def 1; then
U2: ((Omega-->K)-RV).w = (Omega-->K).w + ((-1)(#)RV).w by VALUED_1:def 1;
    dom ((-1)(#)RV)=Omega by FUNCT_2:def 1; then
u3: ((Omega-->K)-RV).w = (Omega-->K).w + ((-1)*RV.w) by U2,VALUED_1:def 5;
L1: RV.w + - (Omega-->K).w = RV.w + (-1)*(Omega-->K).w;
   dom ((-1)(#)(Omega-->K)) = Omega by FUNCT_2:def 1;
   then
L2: RV.w + - (Omega-->K).w = RV.w+((-1)(#)(Omega-->K)).w
    by L1,VALUED_1:def 5;
    w in dom RV /\ dom (-(Omega-->K)) by SS; then
   w in dom(RV+ (-(Omega-->K))) by VALUED_1:def 1;
   then RV.w + - (Omega-->K).w = (RV + (-(Omega-->K))).w by L2,VALUED_1:def 1;
 then -(RV-(Omega-->K)).w =((Omega-->K)-RV).w by u3;
 hence thesis by B40,S1,COMPLEX1:70;
 end;
 suppose S1: (RV-(Omega-->K)).w=0;
   0 = Straddle(RV,K).w
   proof
H2: RV.w + ((-1)(#)(Omega-->K)).w=0 by VALUED_1:1,S1;
    dom(Omega-->K)=Omega & dom ((-1)(#)(Omega-->K)) = Omega
      by FUNCT_2:def 1; then
h1: RV.w + (-1)*(Omega-->K).w=0 by H2,VALUED_1:def 5;
    dom ( (-1)(#)RV ) = Omega by FUNCT_2:def 1; then
G2: (Omega-->K).w + ((-1)(#)RV).w=0 by h1,VALUED_1:def 5;
    dom (Omega-->K)=Omega & dom (-RV)=Omega by FUNCT_2:def 1;
    then w in dom(Omega-->K)/\dom(-RV); then
    w in dom ((Omega-->K) + (-RV)) by VALUED_1:def 1; then
    ((Omega-->K)-RV).w=0 by G2,VALUED_1:def 1; then
    Put-Option(RV,K).w=-(RV-(Omega-->K)).w by S1,Def30; then
F3: Call-Option(RV,K).w + Put-Option(RV,K).w = 0 by S1,FINANCE3:def 5;
    dom Call-Option(RV,K) = Omega & dom Put-Option(RV,K) = Omega
     by FUNCT_2:def 1;
    then w in dom Call-Option(RV,K) /\ dom Put-Option(RV,K); then
    w in dom (Call-Option(RV,K) + Put-Option(RV,K)) by VALUED_1:def 1;
    hence thesis by F3,VALUED_1:def 1;
   end;
  hence thesis by COMPLEX1:43,S1;
 end;
end;
