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 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;
