theorem Th111:
  ||.x - w.|| <= ||.x - y.|| + ||.y - w.||
proof
  x - w = x + (09(CNS) + (-w)) by RLVECT_1:4
    .= x + (((-y) + y) + (-w)) by RLVECT_1:5
    .= x + ((-y) + (y + (-w))) by RLVECT_1:def 3
    .= (x - y) + (y - w) by RLVECT_1:def 3;
  hence thesis by Def13;
end;
