let P be set ; :: thesis: for m1, m2, m3 being marking of P st m1 c= m2 & m2 c= m3 holds
m3 - m2 c= m3 - m1
let m1, m2, m3 be marking of P; :: thesis: ( m1 c= m2 & m2 c= m3 implies m3 - m2 c= m3 - m1 )
assume A1: ( m1 c= m2 & m2 c= m3 ) ; :: thesis: m3 - m2 c= m3 - m1
then A2: m1 c= m3 by Th14;
let p be set ; :: according to PNPROC_1:def 4 :: 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, Def4;
A5: (m3 - m1) . p = (m3 . p) - (m1 . p) by A2, A3, Def6;
(m3 - m2) . p = (m3 . p) - (m2 . p) by A1, A3, Def6;
hence p multitude_of <= p multitude_of by A4, A5, XREAL_1:12; :: thesis: verum