let P be set ; :: thesis: for N being Petri_net of P
for R being process of N holds (NeutralProcess N) before R = R
let N be Petri_net of P; :: thesis: for R being process of N holds (NeutralProcess N) before R = R
let R be process of N; :: thesis: (NeutralProcess N) before R = R
thus (NeutralProcess N) before R c= R :: according to XBOOLE_0:def 10 :: thesis: R c= (NeutralProcess N) before R
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (NeutralProcess N) before R or x in R )
assume x in (NeutralProcess N) before R ; :: thesis: x in R
then consider C1, C being firing-sequence of N such that
A1: x = C1 ^ C and
A2: C1 in NeutralProcess N and
A3: C in R ;
C1 = <*> N by A2, TARSKI:def 1;
hence x in R by A1, A3, FINSEQ_1:34; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in R or x in (NeutralProcess N) before R )
assume A4: x in R ; :: thesis: x in (NeutralProcess N) before R
then reconsider C = x as Element of N * ;
A5: <*> N in NeutralProcess N by TARSKI:def 1;
x = (<*> N) ^ C by FINSEQ_1:34;
hence x in (NeutralProcess N) before R by A4, A5; :: thesis: verum