reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th7:
  for f be Functional_Sequence of X,ExtREAL holds
  for x be Element of X st x in dom (f.0) holds
  (inferior_realsequence f)#x = inferior_realsequence(f#x)
proof
  let f be Functional_Sequence of X,ExtREAL;
  set F = inferior_realsequence f;
  hereby
    let x be Element of X;
    assume
A1: x in dom (f.0);
    now
      let n be Element of NAT;
A2:   (F#x).n = (F.n).x by MESFUNC5:def 13;
      dom(F.n) = dom (f.0) by Def5;
      hence (F#x).n =(inferior_realsequence (f#x)).n by A1,A2,Def5;
    end;
    hence F#x = inferior_realsequence(f#x) by FUNCT_2:63;
  end;
end;
