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> = - <i> * <k>
proof
A1: <i> = [*zz,jj,zz,zz*] by Lm2,Lm3;
  then
A2: <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
    .= [*zz,zz,jj,zz*];
  <i> * <k> = [* 0*0-1*0-0*0-0*1, 0*0+1*0+0*1-0*0,
  0*0+0*0+0*1-1*1, 0*1+0*0+1*0-0*0 *] by A1,Def9
    .= [*zz,zz,-jj,zz*];
  then <k> * <i> + <i> * <k> = [*0+0,0+0,1+ -1,0+0*] by A2,Def6
    .= 0 by Lm6;
  hence thesis by Def7;
end;
