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+;

theorem
  <k> * <k> = -1
proof
A1: <k> * <k> = [* 0*0-0*0-0*0-1*1, 0*0+0*0+0*1-1*0,
  0*0+0*0+0*1-1*0, 0*1+1*0+0*0-0*0 *] by Def9
    .= [*-1,0,0,0*];
  -1 = [*-jj,zz*] by ARYTM_0:def 5;
  hence thesis by A1,Lm3;
end;
