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 Th35:
  {<*e1*>} concur {<*e2*>} = {<*e1,e2*>, <*e2,e1*>}
proof
  set C1 = <*e1*>, C2 = <*e2*>;
  set R1 = {C1}, R2 = {C2};
  thus {C1} concur {C2} c= {<*e1,e2*>, <*e2,e1*>}
  proof
    let x be object;
    assume x in {C1} concur {C2};
    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 & Seq q2 in R2;
    consider q1,q2 being FinSubsequence such that
A3: C = q1 \/ q2 and
A4: q1 misses q2 and
A5: Seq q1 in R1 and
A6: Seq q2 in R2 by A2;
A7: Seq q1 = C1 by A5,TARSKI:def 1;
A8: Seq q2 = C2 by A6,TARSKI:def 1;
    consider i being Element of NAT such that
A9: q1 = {[i,e1]} by A7,FINSEQ_3:159;
    consider j being Element of NAT such that
A10: q2 = {[j,e2]} by A8,FINSEQ_3:159;
A11: [i,e1] in q1 by A9,TARSKI:def 1;
A12: [j,e2] in q2 by A10,TARSKI:def 1;
A13: C = {[i,e1], [j,e2]} by A3,A9,A10,ENUMSET1:1;
    then i = 1 & j = 1 & e1 = e2 or i = 1 & j = 2 or i = 2 & j = 1
            by FINSEQ_1:98;
    then C = <*e1,e2*> or C = <*e2,e1*>
       by A4,A11,A12,A13,FINSEQ_1:99,XBOOLE_0:3;
    hence thesis by A1,TARSKI:def 2;
  end;
  let x be object;
  assume
A14: x in {<*e1,e2*>, <*e2,e1*>};
  per cases by A14,TARSKI:def 2;
  suppose
A15: x = <*e1,e2*>;
    then
A16: x = {[1,e1], [2,e2]} by FINSEQ_1:99
      .= {[1,e1]} \/ {[2,e2]} by ENUMSET1:1;
    reconsider q1 = {[1,e1]}, q2 = {[2,e2]} as FinSubsequence by FINSEQ_1:96;
    [1,e1] <> [2,e2] by XTUPLE_0:1;
    then not [1,e1] in q2 by TARSKI:def 1;
    then
A17: q1 misses q2 by ZFMISC_1:50;
A18: Seq q1 = <*e1*> by FINSEQ_3:157;
A19: Seq q2 = <*e2*> by FINSEQ_3:157;
A20: <*e1*> in R1 by TARSKI:def 1;
    <*e2*> in R2 by TARSKI:def 1;
    hence thesis by A15,A16,A17,A18,A19,A20;
  end;
  suppose
A21: x = <*e2,e1*>;
    then
A22: x = {[1,e2], [2,e1]} by FINSEQ_1:99
      .= {[1,e2]} \/ {[2,e1]} by ENUMSET1:1;
    reconsider q1 = {[2,e1]}, q2 = {[1,e2]} as FinSubsequence by FINSEQ_1:96;
    [1,e2] <> [2,e1] by XTUPLE_0:1;
    then not [2,e1] in q2 by TARSKI:def 1;
    then
A23: q1 misses q2 by ZFMISC_1:50;
A24: Seq q1 = <*e1*> by FINSEQ_3:157;
A25: Seq q2 = <*e2*> by FINSEQ_3:157;
A26: <*e1*> in R1 by TARSKI:def 1;
    <*e2*> in R2 by TARSKI:def 1;
    hence thesis by A21,A22,A23,A24,A25,A26;
  end;
end;
