reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem Th1:
  for X be non empty set, f be Functional_Sequence of X,REAL, x be
  Element of X holds f#x = (R_EAL f)#x
proof
  let X be non empty set;
  let f be Functional_Sequence of X,REAL;
  let x be Element of X;
  now
    let r be object;
    assume r in rng((R_EAL f)#x);
    then consider n be object such that
A1: n in NAT and
A2: ((R_EAL f)#x).n = r by FUNCT_2:11;
    reconsider n as Element of NAT by A1;
    r = ((R_EAL f).n).x by A2,MESFUNC5:def 13
      .= (R_EAL(f.n)).x
      .= (f.n).x;
    hence r in REAL by XREAL_0:def 1;
  end;
  then rng((R_EAL f)#x) c= REAL;
  then reconsider RFx = (R_EAL f)#x as sequence of REAL by FUNCT_2:6;
  reconsider RFx as Real_Sequence;
  now
    let n be object;
    assume n in NAT;
    then reconsider n1 = n as Element of NAT;
    RFx.n = ((R_EAL f).n1).x by MESFUNC5:def 13
      .= (R_EAL(f.n1)).x;
    hence RFx.n = (f#x).n by SEQFUNC:def 10;
  end;
  hence thesis by FUNCT_2:12;
end;
