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 Th31:
  {C1} before {C2} = {C1^C2}
proof
  thus {C1} before {C2} c= {C1^C2}
  proof
    let x be object;
    assume x in {C1} before {C2};
    then consider fs1, fs2 such that
A1: x = fs1^fs2 and
A2: fs1 in {C1} and
A3: fs2 in {C2};
A4: fs1 = C1 by A2,TARSKI:def 1;
    fs2 = C2 by A3,TARSKI:def 1;
    hence thesis by A1,A4,TARSKI:def 1;
  end;
  let x be object;
  assume x in {C1^C2};
  then
A5: x = C1^C2 by TARSKI:def 1;
A6: C1 in {C1} by TARSKI:def 1;
  C2 in {C2} by TARSKI:def 1;
  hence thesis by A5,A6;
end;
