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 Th67:
  for a,b,c,d being Real holds a^2 + b^2 + c^2 + d^2 >= 0
proof
  let a,b,c,d be Real;
A1: a^2 >= 0 by XREAL_1:63;
A2: b^2>= 0 by XREAL_1:63;
A3: c^2 >= 0 by XREAL_1:63;
  d^2 >= 0 by XREAL_1:63;
  hence thesis by A1,A2,A3;
end;
