reserve q,r,c,c1,c2,c3 for Quaternion;
reserve x1,x2,x3,x4,y1,y2,y3,y4 for Real;

theorem Th19:
  -c = (-1q) * c
proof
  consider x,y,w,z be Element of REAL such that
A1: c = [*x,y,w,z*] by Lm1;
A2: c + [*-x,-y,-w,-z*] = [*x + -x,y + -y,w + -w,z + -z*] by A1,QUATERNI:def 7
    .= [*In(0,REAL),In(0,REAL)*] by QUATERNI:91
    .= 0 by ARYTM_0:def 5;
  1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*1,0,0,0*] by QUATERNI:91;
  then 1q + [*-1,0,0,0*] =[*jj+ -jj,0+0,0+0,0+0*] by QUATERNI:def 7
    .=[*In(0,REAL),In(0,REAL)*] by QUATERNI:91
    .=0 by ARYTM_0:def 5;
  then (-1q)*c = [*-1,0,0,0*]*[*x,y,w,z*] by A1,QUATERNI:def 8
    .= [* (-1)*x-0*y-0*w-0*z,
  (-1)*y+0*x+0*z-0*w, (-1)*w+x*0+y*0-z*0,
  (-jj)*z+0*x+0*w-0*y *] by QUATERNI:def 10;
  hence thesis by A2,QUATERNI:def 8;
end;
