reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;

theorem Th8:
  A in F implies (A,{}X) followed_by {}X is Covering of A,F
proof
  reconsider Sets = (A,{}X) followed_by {}X as SetSequence of X;
A1: Sets.0 = A by FUNCT_7:122;
A2: Sets.1 = {}X by FUNCT_7:123;
  assume
A3: A in F;
  for n be Nat holds Sets.n in F
  proof
    let n be Nat;
    per cases by NAT_1:25;
    suppose
      n = 0;
      hence Sets.n in F by A3,FUNCT_7:122;
    end;
    suppose
      n = 1;
      hence Sets.n in F by A2,PROB_1:4;
    end;
    suppose
      n > 1;
      then Sets.n = {} by FUNCT_7:124;
      hence Sets.n in F by PROB_1:4;
    end;
  end;
  then reconsider Sets as Set_Sequence of F by Def2;
  A c= union rng Sets
  proof
    let x be object;
    dom Sets = NAT by FUNCT_2:def 1;
    then
A4: Sets.0 in rng Sets by FUNCT_1:3;
    assume x in A;
    hence x in union rng Sets by A1,A4,TARSKI:def 4;
  end;
  hence (A,{}X) followed_by {}X is Covering of A,F by Def3;
end;
