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 holds A3.n = A1.n \/ A2.n) implies for n holds (
  superior_setsequence(A3)).n = (superior_setsequence(A1)).n \/ (
  superior_setsequence(A2)).n
proof
  assume
A1: for n holds A3.n = A1.n \/ A2.n;
  let n;
  reconsider X3 = superior_setsequence A3 as SetSequence of X;
  reconsider X2 = superior_setsequence A2 as SetSequence of X;
  set B = A1;
  reconsider X1 = superior_setsequence B as SetSequence of X;
A2: X1.n \/ X2.n c= X3.n
  proof
    let x be object;
    assume
A3: x in (X1.n \/ X2.n);
      per cases by A3,XBOOLE_0:def 3;
      suppose
        x in X1.n;
        then consider k being Nat such that
A4:     x in B.(n+k) by Th20;
        A3.(n+k) = B.(n+k) \/ A2.(n+k) by A1;
        then x in A3.(n+k) by A4,XBOOLE_0:def 3;
        hence thesis by Th20;
      end;
      suppose
        x in X2.n;
        then consider k being Nat such that
A5:     x in A2.(n+k) by Th20;
        A3.(n+k) = B.(n+k) \/ A2.(n+k) by A1;
        then x in A3.(n+k) by A5,XBOOLE_0:def 3;
        hence thesis by Th20;
      end;
  end;
  X3.n c= X1.n \/ X2.n
  proof
    let x be object;
    assume x in X3.n;
    then consider k being Nat  such that
A6: x in A3.(n+k) by Th20;
A7: x in (B.(n+k) \/ A2.(n+k)) by A1,A6;
    now
      per cases by A7,XBOOLE_0:def 3;
      case
        x in B.(n+k);
        hence x in X1.n by Th20;
      end;
      case
        x in A2.(n+k);
        hence x in X2.n by Th20;
      end;
    end;
    hence thesis by XBOOLE_0:def 3;
  end;
  hence thesis by A2,XBOOLE_0:def 10;
end;
