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;

theorem Th27:
  ex f st (dom f = Sigma & for D st D in Sigma holds (p in D
  implies f.D = 1) & (not p in D implies f.D = 0))
proof
  deffunc G(set) = 0;
  deffunc F(set) = 1;
  defpred C[set] means p in $1;
  ex f being Function st dom f = Sigma & for x being set st x in Sigma
holds (C[x] implies f.x=F(x)) & (not C[x] implies f.x=G(x))
    from PARTFUN1:sch 5;
  then consider f being Function such that
A1: dom f = Sigma and
A2: for x being set st x in Sigma holds (C[x] implies f.x = 1) & (not C[
  x] implies f.x = 0);
  take f;
  thus dom f = Sigma by A1;
  let D;
  assume
A3: D in Sigma;
  hence p in D implies f.D = 1 by A2;
  assume not p in D;
  hence thesis by A2,A3;
end;
