defpred S_{1}[ object ] means ( p in loc D & $1 = ColD . p );

deffunc H_{1}( object ) -> Element of NAT = ((m . p) . (In ($1, the ColoredSet of CPN))) + 1;

deffunc H_{2}( object ) -> Element of NAT = (m . p) . (In ($1, the ColoredSet of CPN));

P1: for x being object st x in the ColoredSet of CPN holds

( ( S_{1}[x] implies H_{1}(x) in NAT ) & ( not S_{1}[x] implies H_{2}(x) in NAT ) )
;

consider f being Function of the ColoredSet of CPN,NAT such that

P2: for x being object st x in the ColoredSet of CPN holds

( ( S_{1}[x] implies f . x = H_{1}(x) ) & ( not S_{1}[x] implies f . x = H_{2}(x) ) )
from FUNCT_2:sch 5(P1);

take f ; :: thesis: for x being Element of the ColoredSet of CPN holds

( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) )

thus for x being Element of the ColoredSet of CPN holds

( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) ) :: thesis: verum

deffunc H

deffunc H

P1: for x being object st x in the ColoredSet of CPN holds

( ( S

consider f being Function of the ColoredSet of CPN,NAT such that

P2: for x being object st x in the ColoredSet of CPN holds

( ( S

take f ; :: thesis: for x being Element of the ColoredSet of CPN holds

( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) )

thus for x being Element of the ColoredSet of CPN holds

( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) ) :: thesis: verum

proof

let x be Element of the ColoredSet of CPN; :: thesis: ( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) )

( ( S_{1}[x] implies f . x = H_{1}(x) ) & ( not S_{1}[x] implies f . x = H_{2}(x) ) )
by P2;

hence ( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) ) ; :: thesis: verum

end;( ( S

hence ( ( p in loc D & x = ColD . p implies f . x = ((m . p) . x) + 1 ) & ( ( not p in loc D or not x = ColD . p ) implies f . x = (m . p) . x ) ) ; :: thesis: verum