 reserve Omega for non empty set;
 reserve F for SigmaField of Omega;
 reserve phi for Real_Sequence;
 reserve jpi for pricefunction;

theorem
  ex G being sequence of
     set_of_random_variables_on(Special_SigmaField1,Borel_Sets) st
  (G.0={1,2,3,4}-->1 & G.1={1,2,3,4}-->5 &
  for k being Nat st k>1 holds G.k={1,2,3,4}-->0)
proof
 deffunc U(Nat)=IFEQ ($1,0,RVfirst (1),IFEQ ($1,1,RVfirst(5),RVfirst(0)));
 consider f being sequence of
  set_of_random_variables_on(Special_SigmaField1,Borel_Sets) such that
  A1: for d being Element of NAT holds f.d=U(d) from FUNCT_2:sch 4;
 take f;
b1:f.0=IFEQ (0,0,RVfirst (1),IFEQ (0,1,RVfirst(5),RVfirst(0))) by A1;
  f.1=IFEQ (1,0,RVfirst (1),IFEQ (1,1,RVfirst(5),RVfirst(0))) by A1;
  then b2: f.1 = IFEQ (1,1,RVfirst(5),RVfirst(0)) by FUNCOP_1:def 8;
  for k being Nat st k>1 holds f.k={1,2,3,4}-->0
  proof
   let k be Nat;
   assume C1: k>1;
   k in NAT by ORDINAL1:def 12;
   then f.k=IFEQ (k,0,RVfirst (1),IFEQ (k,1,RVfirst(5),RVfirst(0))) by A1;
   then f.k=IFEQ (k,1,RVfirst(5),RVfirst(0)) by C1,FUNCOP_1:def 8;
  hence thesis by C1,FUNCOP_1:def 8;
  end;
 hence thesis by b1,b2,FUNCOP_1:def 8;
end;
