reserve i, j, k, l, m, n for Nat,
  a, b, c, t, u for object,
  X, Y, Z for set,
  D, D1, D2, Fml for non empty set;
reserve p, q, r, s for FinSequence;
 reserve R, R1, R2 for Rule;
 reserve A, A1, A2 for non empty set;
 reserve B, B1, B2 for set;
 reserve P, P1, P2 for Formula-sequence;
 reserve S, S1, S2 for Formula-finset;
 reserve C for Extension of B;
 reserve E for Extension of R;

theorem
  for B, R, R1, R2 holds R1 \/ R2 is (B,R)-derivable
      iff R1 is (B,R)-derivable & R2 is (B,R)-derivable
proof
  let B, R, R1, R2;
  thus R1 \/ R2 is (B,R)-derivable
      implies R1 is (B,R)-derivable & R2 is (B,R)-derivable
  proof
    assume A1: R1 \/ R2 is (B,R)-derivable;
    thus R1 is (B,R)-derivable
      proof
      let S, t;
      assume [S, t] in R1;
      then [S, t] in R1 \/ R2 by XBOOLE_0:def 3;
      hence thesis by A1;
      end;
    let S, t;
    assume [S, t] in R2;
    then [S, t] in R1 \/ R2 by XBOOLE_0:def 3;
    hence thesis by A1;
    end;
  assume that
    A10: R1 is (B,R)-derivable and
    A11: R2 is (B,R)-derivable;
  let S, t;
  assume [S, t] in R1 \/ R2;
  then per cases by XBOOLE_0:def 3;
    suppose [S, t] in R1;
    hence thesis by A10;
    end;
    suppose [S, t] in R2;
    hence thesis by A11;
  end;
end;
