reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem
  (for n being Nat holds A3.n = A1.n \/ A2.n) implies
   for n being Nat holds (
  inferior_setsequence A1).n \/ (inferior_setsequence(A2)).n c= (
  inferior_setsequence(A3)).n
proof
  assume
A1: for n being Nat holds A3.n = A1.n \/ A2.n;
  let n be Nat;
  set X1 = inferior_setsequence A1, X2 = inferior_setsequence A2, X3 =
  inferior_setsequence A3;
  now
    let x be object;
    assume
A2: x in X1.n \/ X2.n;
    per cases by A2,XBOOLE_0:def 3;
    suppose
A3:   x in X1.n;
      now
        let k be Nat;
         x in A1.(n+k) & A3.(n+k) = A1.(n+k) \/ A2.(n+k) by A1,A3,Th19;
        hence x in A3.(n+k) by XBOOLE_0:def 3;
      end;
      hence x in X3.n by Th19;
    end;
    suppose
A4:   x in X2.n;
      now
        let k be Nat;
        x in A2.(n+k) & A3.(n+k) = A1.(n+k) \/ A2.(n+k) by A1,A4,Th19;
        hence x in A3.(n+k) by XBOOLE_0:def 3;
      end;
      hence x in X3.n by Th19;
    end;
  end;
  hence thesis;
end;
