 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;
 reserve kap for RIF of R;

theorem :: Proposition 7 c1)
  (delta_1 R).(X,Y) + (delta_1 R).(Y,Z) >= (delta_1 R).(X,Z)
  proof
    set m1 = (CMap kappa_1 R).(X,Y), m2 = (CMap kappa_1 R).(Y,Z);
    set m3 = (CMap kappa_1 R).(X,Z), m4 = (CMap kappa_1 R).(Z,Y),
        m5 = (CMap kappa_1 R).(Y,X), m6 = (CMap kappa_1 R).(Z,X);
A1: m1 + m2 >= m3 by Prop6d1;
    m4 + m5 >= m6 by Prop6d1; then
    m1 + m2 + (m4 + m5) >= m3 + m6 by A1,XREAL_1:7; then
    m1 + m5 + (m2 + m4) >= m3 + m6; then
    (delta_1 R).(X,Y) + (m2 + m4) >= m3 + m6 by Delta1; then
    (delta_1 R).(X,Y) + (delta_1 R).(Y,Z) >= m3 + m6 by Delta1;
    hence thesis by Delta1;
  end;
