theorem Th8:
  m c= m1 & m1 c= m2 implies m1 - m c= m2 - m
proof
  assume
A1: m c= m1;
  assume
A2: m1 c= m2;
  let p;
  assume
A3: p in P;
A4: m c= m2 by A1,A2,Th2;
  m1.p <= m2.p by A2,A3;
  then (m1.p - m.p) <= (m2.p - m.p) by XREAL_1:9;
  then (m1 - m).p <= m2.p - m.p by A1,A3,Def5;
  hence thesis by A3,A4,Def5;
end;
