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 Th29:
  P * ASeq is Real_Sequence
proof
  rng ASeq c= Sigma by RELAT_1:def 19;
  then rng ASeq c= dom P by FUNCT_2:def 1;
  then
A1: dom (P * ASeq) = dom ASeq by RELAT_1:27
    .= NAT by FUNCT_2:def 1;
  rng (P * ASeq) c= REAL by RELAT_1:def 19;
  hence thesis by A1,FUNCT_2:def 1,RELSET_1:4;
end;
