let P be set ; :: thesis: for m1, m2, m3 being marking of P st m1 c= m2 & m2 c= m3 holds
m1 c= m3
let m1, m2, m3 be marking of P; :: thesis: ( m1 c= m2 & m2 c= m3 implies m1 c= m3 )
assume that
A1: m1 c= m2 and
A2: m2 c= m3 ; :: thesis: m1 c= m3
let p be set ; :: according to PNPROC_1:def 3 :: thesis: ( p in P implies p multitude_of <= p multitude_of )
assume A3: p in P ; :: thesis: p multitude_of <= p multitude_of
then A4: m1 . p <= m2 . p by A1;
m2 . p <= m3 . p by A2, A3;
hence p multitude_of <= p multitude_of by A4, XXREAL_0:2; :: thesis: verum