reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;
reserve B,C for Coset of W;
reserve CNS for ComplexNormSpace;
reserve x, y, w, g, g1, g2 for Point of CNS;

theorem Th110:
  |.||.x.|| - ||.y.||.| <= ||.x - y.||
proof
  (y - x) + x = y - (x - x) by RLVECT_1:29
    .= y - 09(CNS) by RLVECT_1:15
    .= y by RLVECT_1:13;
  then ||.y.|| <= ||.y - x.|| + ||.x.|| by Def13;
  then ||.y.|| - ||.x.|| <= ||.y - x.|| by XREAL_1:20;
  then ||.y.|| - ||.x.|| <= ||.x - y.|| by Th108;
  then
A1: - ||.x - y.|| <= -(||.y.|| - ||.x.||) by XREAL_1:24;
  ||.x.|| - ||.y.|| <= ||.x - y.|| by Th109;
  hence thesis by A1,ABSVALUE:5;
end;
