reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;
reserve F for Field_Subset of X;
reserve ASeq,BSeq for SetSequence of Omega;
reserve A1 for SetSequence of X;
reserve Sigma for SigmaField of Omega;
reserve Si for SigmaField of X;
reserve A, B for Event of Sigma,
  ASeq for SetSequence of Sigma;
reserve P for Function of Sigma,REAL;

theorem Th28:
  ex P st for D st D in Sigma holds (p in D implies P.D = 1) & (
  not p in D implies P.D = 0)
proof
  consider f such that
A1: dom f = Sigma and
A2: for D st D in Sigma holds (p in D implies f.D = 1) & (not p in D
  implies f.D = 0) by Th27;
A3: 0 in REAL by XREAL_0:def 1;
  rng f c= REAL
  proof
    let y be object;
    assume y in rng f;
    then consider x being object such that
A4: x in dom f and
A5: y = f.x by FUNCT_1:def 3;
    reconsider x as Subset of Omega by A1,A4;
    reconsider j = 1 as Real;
A6: 1 in REAL by XREAL_0:def 1;
A7: not p in x implies y = 0 by A1,A2,A4,A5;
    p in x implies y = j by A1,A2,A4,A5;
    hence thesis by A7,A3,A6;
  end;
  then reconsider f as Function of Sigma,REAL by A1,FUNCT_2:def 1,RELSET_1:4;
  take f;
  thus thesis by A2;
end;
