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> * <i> = <j>
proof
  <i> = [*0,1,0,0*] by Lm2,Lm3;
  then <k> * <i> = [* 0*0-0*1-0*0-1*0, 0*1+0*0+0*0-1*0,
  0*0+0*0+1*1-0*0, 0*0+1*0+0*0-0*1 *] by Def9
    .= [*0,0,1,0*];
  hence thesis;
end;
