reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th18:
  F is with_the_same_dom & (for n be Nat holds G.n = (F.n)|D)
  implies G is with_the_same_dom
proof
  assume that
A1: F is with_the_same_dom and
A2: for n be Nat holds G.n = (F.n)|D;
  let n,m be Nat;
  G.m = (F.m)|D by A2;
  then
A3: dom(G.m) = dom(F.m) /\ D by RELAT_1:61;
  G.n = (F.n)|D by A2;
  then dom(G.n) = dom(F.n) /\ D by RELAT_1:61;
  hence thesis by A1,A3;
end;
