reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;

theorem Th3:
  |.|.q1.|- |.q2.|.|<=|.q1-q2.|
proof
  per cases;
  suppose
    |.q1.|>=|.q2.|;
    then |.q1.|- |.q2.|>=0 by XREAL_1:48;
    then |.q1.|- |.q2.|=|.|.q1.|- |.q2.|.| by ABSVALUE:def 1;
    hence thesis by TOPRNS_1:32;
  end;
  suppose
A1: |.q1.|<|.q2.|;
A2: |.q2.|- |.q1.|<= |.q2-q1.| by TOPRNS_1:32;
    |.q2.|- |.q1.|>0 by A1,XREAL_1:50;
    then |.|.q2.|- |.q1.|.|<= |.q2-q1.| by A2,ABSVALUE:def 1;
    then |.|.q2.|- |.q1.|.|<= |.q1-q2.| by TOPRNS_1:27;
    hence thesis by UNIFORM1:11;
  end;
end;
