set PLA = {0};
set a = 1;
set b = 2;
set TRA = {1};
set TSA = [:{1},{0}:];
[:{1},{0}:] c= [:{1},{0}:] ;
then reconsider TSA = [:{1},{0}:] as non empty Relation of {1},{0} ;
set STA = [:{0},{1}:];
[:{0},{1}:] c= [:{0},{1}:] ;
then reconsider STA = [:{0},{1}:] as non empty Relation of {0},{1} ;
set CS = {1,2};
set CS0 = {1};
set fa = the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}));
set f = {1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}));
set CPNT = Colored_PT_net_Str(# {0},{1},STA,TSA,{1,2},({1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}))) #);
A1: Colored_PT_net_Str(# {0},{1},STA,TSA,{1,2},({1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}))) #) is with_S-T_arc ;
Colored_PT_net_Str(# {0},{1},STA,TSA,{1,2},({1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}))) #) is with_T-S_arc ;
then reconsider CPNT = Colored_PT_net_Str(# {0},{1},STA,TSA,{1,2},({1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}))) #) as Colored_Petri_net by A1;
a0: ( 1 in {1,2} & 2 in {1,2} & 1 <> 2 ) by TARSKI:def 2;

A2: now :: thesis: for t being transition of CPNT st t in dom the firing-rule of CPNT holds
ex CS1 being non empty Subset of the ColoredSet of CPNT ex I being Subset of (*' {t}) ex O being Subset of ({t} *') st the firing-rule of CPNT . t is Function of (thin_cylinders (CS1,I)),(thin_cylinders (CS1,O))

{1} c= {1,2}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {1} or x in {1,2} )
assume x in {1} ; :: thesis: x in {1,2}
then x = 1 by TARSKI:def 1;
hence x in {1,2} by TARSKI:def 2; :: thesis: verum

end;

then reconsider CS1 = {1} as non empty Subset of the ColoredSet of CPNT ;
let t be transition of CPNT; :: thesis: ( t in dom the firing-rule of CPNT implies ex CS1 being non empty Subset of the ColoredSet of CPNT ex I being Subset of (*' {t}) ex O being Subset of ({t} *') st the firing-rule of CPNT . t is Function of (thin_cylinders (CS1,I)),(thin_cylinders (CS1,O)) )
assume t in dom the firing-rule of CPNT ; :: thesis: ex CS1 being non empty Subset of the ColoredSet of CPNT ex I being Subset of (*' {t}) ex O being Subset of ({t} *') st the firing-rule of CPNT . t is Function of (thin_cylinders (CS1,I)),(thin_cylinders (CS1,O))
A3: t = 1 by TARSKI:def 1;
A4: 1 in {1} by TARSKI:def 1;
A5: 0 in {0} by TARSKI:def 1;
then [1,0] in TSA by A4, ZFMISC_1:87;
then reconsider O = {0} as Subset of ({t} *') by ZFMISC_1:31, A3, PETRI:8;
[0,1] in STA by A5, A4, ZFMISC_1:87;
then reconsider I = {0} as Subset of (*' {t}) by ZFMISC_1:31, A3, PETRI:6;
A6: the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0})) is Function of (thin_cylinders (CS1,I)),(thin_cylinders (CS1,O)) ;
({1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}))) . t = the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0})) by FUNCOP_1:7;
hence ex CS1 being non empty Subset of the ColoredSet of CPNT ex I being Subset of (*' {t}) ex O being Subset of ({t} *') st the firing-rule of CPNT . t is Function of (thin_cylinders (CS1,I)),(thin_cylinders (CS1,O)) by A6; :: thesis: verum

end;

A7: dom ({1} --> the Function of (thin_cylinders ({1},{0})),(thin_cylinders ({1},{0}))) = {1} by FUNCOP_1:13;

now :: thesis: for x being object st x in dom the firing-rule of CPNT holds
x in the carrier' of CPNT \ (Outbds CPNT)

reconsider a1 = 1 as transition of CPNT by TARSKI:def 1;
let x be object ; :: thesis: ( x in dom the firing-rule of CPNT implies x in the carrier' of CPNT \ (Outbds CPNT) )
0 in {0} by TARSKI:def 1;
then a8: [a1,0] in TSA by ZFMISC_1:87;
A9: not a1 in Outbds CPNT

proof

assume a1 in Outbds CPNT ; :: thesis: contradiction
then ex x being transition of CPNT st
( a1 = x & x is outbound ) ;
hence contradiction by a8, PETRI:8; :: thesis: verum

end;

assume x in dom the firing-rule of CPNT ; :: thesis: x in the carrier' of CPNT \ (Outbds CPNT)
then x = 1 by A7, TARSKI:def 1;
hence x in the carrier' of CPNT \ (Outbds CPNT) by A9, XBOOLE_0:def 5; :: thesis: verum

end;

then dom the firing-rule of CPNT c= the carrier' of CPNT \ (Outbds CPNT) ;
then reconsider CPNT = CPNT as strict Colored-PT-net-like Colored_Petri_net by A2, PETRI_2:def 11;
take CPNT ; :: thesis: CPNT is with-nontrivial-ColoredSet
thus CPNT is with-nontrivial-ColoredSet by a0, ZFMISC_1:def 10; :: thesis: verum