reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem
  |.w1 - w2.| <= |.w1 - w.| + |.w - w2.|
proof
  0.TOP-REAL N = w - w by RLVECT_1:5
    .= -w + w;
  then |.w1 - w2.| = |.w1 + ((-w) + w) - w2.| by RLVECT_1:4
    .= |.w1 + (-w) + w - w2.| by RLVECT_1:def 3
    .= |.(w1 - w) + w - w2.|
    .= |.(w1 - w) + (w - w2).| by RLVECT_1:def 3;
  hence thesis by Th29;
end;
