reserve i,j,k,l for Nat,
  x,x1,x2,y1,y2 for set;
reserve P,p,x,y,x1,x2 for set,
  m1,m2,m3,m4,m for marking of P,
  i,j,j1,j2,k,k1,k2,l,l1 for Nat;

theorem Th9:
  m1 c= m2 implies m2 + m3 -m1 = m2 - m1 + m3
proof
  assume
A1: m1 c= m2;
  let p be object;
  assume
A2: p in P;
  m2 c= m2 + m3 by Th4;
  then
A3: m1 c= m2 + m3 by A1,Th2;
  (m2 - m1 + m3).p = (m2 - m1).p + m3.p by A2,Def4
    .= m3.p + (m2.p - m1.p) by A1,A2,Def5
    .= m3.p + m2.p - m1.p
    .= (m3 + m2).p - m1.p by A2,Def4
    .= (m2 + m3 - m1).p by A2,A3,Def5;
  hence thesis;
end;
