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

theorem
  (-c)" = -(c")
proof
A1: 1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*1,0,0,0*] by QUATERNI:91;
  consider x1,x2,x3,x4 be Element of REAL such that
A2: c = [*x1,x2,x3,x4*] by Lm1;
A3: -c = [*-x1,-x2,-x3,-x4*] by A2,Th4;
A4: |.-c.| = |.c.| by QUATERNI:72;
A5: c" = [*(x1*1+x2*0+x3*0+x4*0)/(|.c.|^2), (x1*0-x2*1-x3*0+x4*0)/(|.c.|^2),
  (x1*0+x2*0-x3*1-x4*0)/(|.c.|^2),
  (x1*0-x2*0+x3*0-x4*jj)/(|.c.|^2) *] by A1,A2,Def1,Lm5
    .= [*x1/|.c.|^2,-x2/|.c.|^2,-x3/|.c.|^2,-x4/|.c.|^2 *];
  (-c)" = [*((-x1)*1+(-x2)*0+(-x3)*0+(-x4)*0)/(|.(-c).|^2),
  ((-x1)*0-(-x2)*1-(-x3)*0+(-x4)*0)/(|.(-c).|^2),
  ((-x1)*0+(-x2)*0-(-x3)*1-(-x4)*0)/(|.(-c).|^2),
  ((-x1)*0-(-x2)*0+(-x3)*0-(-x4)*jj)/(|.(-c).|^2) *] by A1,A3,Def1,Lm5
    .= [*-x1/|.c.|^2,x2/|.c.|^2, x3/|.c.|^2,x4/|.c.|^2 *] by A4;
  then c" + (-c)" = [*x1/|.c.|^2 + -x1/|.c.|^2, -x2/|.c.|^2 + x2/|.c.|^2,
  -x3/|.c.|^2 + x3/|.c.|^2, -x4/|.c.|^2 + x4/|.c.|^2 *]
  by A5,QUATERNI:def 7
    .= [*In(0,REAL),In(0,REAL) *] by QUATERNI:91
    .=0 by ARYTM_0:def 5;
  hence thesis by QUATERNI:def 8;
end;
