
theorem SYLM3:
  for CPNT1, CPNT2 be Colored-PT-net,
  O12 be Function of Outbds(CPNT1), the carrier of CPNT2,
  q12 be Function st
  dom q12=Outbds(CPNT1) &
  for t01 be transition of CPNT1 st t01 is outbound
  holds q12.t01 is Function of thin_cylinders(the ColoredSet of CPNT1,
  *'{t01}), thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01)) holds
  q12 in Funcs(Outbds(CPNT1),
  union {Funcs(thin_cylinders(the ColoredSet of CPNT1,*'{t01}),
  thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01)))
  where t01 is transition of CPNT1:t01 is outbound})
proof
  let CPNT1, CPNT2 be Colored-PT-net,
  O12 be Function of Outbds(CPNT1), the carrier of CPNT2,
  q12 be Function;
  assume
P1: dom q12=Outbds(CPNT1);
  assume
P2: for t01 be transition of CPNT1 st t01 is outbound
  holds q12.t01 is
  Function of thin_cylinders(the ColoredSet of CPNT1,
  *'{t01}), thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01));
  now let y be object;
    assume y in rng q12; then
    consider t01 being object such that
P11: t01 in dom q12 & y = q12.t01 by FUNCT_1:def 3;
    reconsider t01 as transition of CPNT1 by P1, P11;
p12: ex x be transition of CPNT1
    st x = t01 & x is outbound by P1,P11;
    q12.t01 is Function of thin_cylinders(the ColoredSet of CPNT1,
    *'{t01}), thin_cylinders(the ColoredSet of CPNT1, Im(O12, t01))
    by P2, p12; then
P13: q12.t01 in Funcs(thin_cylinders(the ColoredSet of CPNT1,
    *'{t01}), thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01)))
    by FUNCT_2:8;
    Funcs(thin_cylinders(the ColoredSet of CPNT1,
    *'{t01}), thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01))) in
    {Funcs(thin_cylinders(the ColoredSet of CPNT1,*'{t01}),
    thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01)))
    where t01 is transition of CPNT1 : t01 is outbound } by p12;
    hence y in
    union {Funcs(thin_cylinders(the ColoredSet of CPNT1,*'{t01}),
    thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01)))
    where t01 is transition of CPNT1 : t01 is outbound }
    by P11, P13, TARSKI:def 4;
  end;
  then rng q12 c= union {
  Funcs(thin_cylinders( the ColoredSet of CPNT1,*'{t01}),
  thin_cylinders(the ColoredSet of CPNT1, Im(O12,t01)))
  where t01 is transition of CPNT1 : t01 is outbound };
  hence thesis by FUNCT_2:def 2, P1;
end;
