reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;
reserve z,z1,z2,z3,z4 for Quaternion;
 reserve x for Real;

theorem
  for a, b, c, d being Real st a^2 + b^2 + c^2 + d^2 = 0 holds
    a = 0 & b = 0 & c = 0 & d = 0 by Lm9;
