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

theorem Th9:
  (-x1)*c = -(x1*c)
proof
  consider x,y,w,z be Element of REAL such that
A1: c = [*x,y,w,z*] by Lm1;
A2: (-x1)*c = [* (-x1)*x,(-x1)*y,(-x1)*w,(-x1)*z *] by A1,QUATERNI:def 21
    .= [* -x1*x,-x1*y,-x1*w,-x1*z *];
A3: -(x1*c) = -[* x1*x,x1*y,x1*w,x1*z *] by A1,QUATERNI:def 21;
  [* x1*x,x1*y,x1*w,x1*z *] + [* -x1*x,-x1*y,-x1*w,-x1*z *]
  =[* x1*x + (-x1*x),x1*y + (-x1*y),x1*w + (-x1*w),x1*z + (-x1*z) *]
  by QUATERNI:def 7
    .=[*In(0,REAL),In(0,REAL)*] by QUATERNI:91
    .=0 by ARYTM_0:def 5;
  hence thesis by A2,A3,QUATERNI:def 8;
end;
