reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th87:
  diff(a,b,o)=o iff a=b
proof
  diff(a,b,o)=o implies a=b
  proof
    assume diff(a,b,o)=o;
    then opposite(a,o)= opposite(b,o) by Def6;
    hence thesis by Th82;
  end;
  hence thesis by Def6;
end;
