reserve i,j,k,l for Nat,
  x,x1,x2,y1,y2 for set;
reserve P,p,x,y,x1,x2 for set,
  m1,m2,m3,m4,m for marking of P,
  i,j,j1,j2,k,k1,k2,l,l1 for Nat;
reserve t,t1,t2 for transition of P;
reserve N for Petri_net of P;
reserve e, e1,e2 for Element of N;
reserve C,C1,C2,C3,fs,fs1,fs2 for firing-sequence of N;
reserve R, R1, R2, R3, P1, P2 for process of N;

theorem Th34:
  (R1 \/ R2) concur R = (R1 concur R) \/ (R2 concur R)
proof
  thus (R1 \/ R2) concur R c= (R1 concur R) \/ (R2 concur R)
  proof
    let x be object;
    assume x in (R1 \/ R2) concur R;
    then consider C such that
A1: x = C and
A2: ex q1,q2 being FinSubsequence st C = q1 \/ q2 & q1 misses q2 & Seq
    q1 in R1 \/ R2 & Seq q2 in R;
    consider q1,q2 being FinSubsequence such that
A3: C = q1 \/ q2 and
A4: q1 misses q2 and
A5: Seq q1 in R1 \/ R2 and
A6: Seq q2 in R by A2;
    Seq q1 in R1 or Seq q1 in R2 by A5,XBOOLE_0:def 3;
    then x in {C1: ex q1,q2 being FinSubsequence st C1 = q1 \/ q2 & q1
    misses q2 & Seq q1 in R1 & Seq q2 in R} or
    x in {C2: ex q1,q2 being FinSubsequence st C2 = q1 \/ q2 & q1 misses q2
    & Seq q1 in R2 & Seq q2 in R} by A1,A3,A4,A6;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A7: x in (R1 concur R) \/ (R2 concur R);
  per cases by A7,XBOOLE_0:def 3;
  suppose x in R1 concur R;
    then consider C such that
A8: x = C and
A9: ex q1,q2 being FinSubsequence st C = q1 \/ q2 & q1 misses q2 & Seq
    q1 in R1 & Seq q2 in R;
    consider q1,q2 being FinSubsequence such that
A10: C = q1 \/ q2 and
A11: q1 misses q2 and
A12: Seq q1 in R1 and
A13: Seq q2 in R by A9;
    Seq q1 in R1 \/ R2 by A12,XBOOLE_0:def 3;
    hence thesis by A8,A10,A11,A13;
  end;
  suppose x in R2 concur R;
    then consider C such that
A14: x = C and
A15: ex q1,q2 being FinSubsequence st C = q1 \/ q2 & q1 misses q2 & Seq
    q1 in R2 & Seq q2 in R;
    consider q1,q2 being FinSubsequence such that
A16: C = q1 \/ q2 and
A17: q1 misses q2 and
A18: Seq q1 in R2 and
A19: Seq q2 in R by A15;
    Seq q1 in R1 \/ R2 by A18,XBOOLE_0:def 3;
    hence thesis by A14,A16,A17,A19;
  end;
end;
