let P be set ; :: thesis: for N being Petri_net of P

for R being process of N holds R before (NeutralProcess N) = R

let N be Petri_net of P; :: thesis: for R being process of N holds R before (NeutralProcess N) = R

let R be process of N; :: thesis: R before (NeutralProcess N) = R

thus R before (NeutralProcess N) c= R :: according to XBOOLE_0:def 10 :: thesis: R c= R before (NeutralProcess N)

assume A4: x in R ; :: thesis: x in R before (NeutralProcess N)

then reconsider C = x as Element of N * ;

A5: <*> N in NeutralProcess N by TARSKI:def 1;

x = C ^ (<*> N) by FINSEQ_1:34;

hence x in R before (NeutralProcess N) by A4, A5; :: thesis: verum

for R being process of N holds R before (NeutralProcess N) = R

let N be Petri_net of P; :: thesis: for R being process of N holds R before (NeutralProcess N) = R

let R be process of N; :: thesis: R before (NeutralProcess N) = R

thus R before (NeutralProcess N) c= R :: according to XBOOLE_0:def 10 :: thesis: R c= R before (NeutralProcess N)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in R or x in R before (NeutralProcess N) )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in R before (NeutralProcess N) or x in R )

assume x in R before (NeutralProcess N) ; :: thesis: x in R

then consider C1, C being firing-sequence of N such that

A1: x = C1 ^ C and

A2: C1 in R and

A3: C in NeutralProcess N ;

C = <*> N by A3, TARSKI:def 1;

hence x in R by A1, A2, FINSEQ_1:34; :: thesis: verum

end;assume x in R before (NeutralProcess N) ; :: thesis: x in R

then consider C1, C being firing-sequence of N such that

A1: x = C1 ^ C and

A2: C1 in R and

A3: C in NeutralProcess N ;

C = <*> N by A3, TARSKI:def 1;

hence x in R by A1, A2, FINSEQ_1:34; :: thesis: verum

assume A4: x in R ; :: thesis: x in R before (NeutralProcess N)

then reconsider C = x as Element of N * ;

A5: <*> N in NeutralProcess N by TARSKI:def 1;

x = C ^ (<*> N) by FINSEQ_1:34;

hence x in R before (NeutralProcess N) by A4, A5; :: thesis: verum