
theorem Th35:
  for X be non empty set, S be SigmaField of X, A be Element of S,
  F be Finite_Sep_Sequence of S, G be FinSequence st dom F = dom G & (for n be
  Nat st n in dom F holds G.n = F.n /\ A) holds G is Finite_Sep_Sequence of S
proof
  let X be non empty set;
  let S be SigmaField of X;
  let A be Element of S;
  let F be Finite_Sep_Sequence of S;
  let G be FinSequence;
  assume that
A1: dom F = dom G and
A2: for n be Nat st n in dom F holds G.n = F.n /\ A;
  rng G c= S
  proof
    let v be object;
    assume v in rng G;
    then consider k be object such that
A3: k in dom G and
A4: v = G.k by FUNCT_1:def 3;
A5: F.k in rng F by A1,A3,FUNCT_1:3;
    G.k = F.k /\ A by A1,A2,A3;
    hence thesis by A4,A5,FINSUB_1:def 2;
  end;
  then reconsider G as FinSequence of S by FINSEQ_1:def 4;
  now
    let i,j be Nat;
    assume that
A6: i in dom G and
A7: j in dom G and
A8: i <> j;
A9: F.i misses F.j by A8,PROB_2:def 2;
A10: G.j = F.j /\ A by A1,A2,A7;
    G.i = F.i /\ A by A1,A2,A6;
    hence G.i misses G.j by A10,A9,XBOOLE_1:76;
  end;
  hence thesis by MESFUNC3:4;
end;
