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+;
reserve z,z1,z2,z3,z4 for Quaternion;
 reserve x for Real;

theorem Th48:
  (-z)*' = -(z*')
proof
A1: z*' = [*Rea z, -Im1 z, -Im2 z, -Im3 z*] by Th36;
  (-z)*' = [*Rea (-z), -Im1 (-z), -Im2 (-z), -Im3 (-z)*] by Th36;
  then (-z)*' = [*-Rea z, -Im1 (-z), -Im2 (-z), -Im3 (-z)*] by Th34
    .= [*-Rea z, - - Im1 z, -Im2 (-z), -Im3 (-z)*] by Th34
    .= [*-Rea z, Im1 z, - - Im2 z, -Im3 (-z)*] by Th34
    .= [*-Rea z, Im1 z, Im2 z, - -Im3 z*] by Th34
    .= [*-Rea z, Im1 z, Im2 z, Im3 z*];
  then z*' + (-z)*' = [*Rea z+ -Rea z, -Im1 z+ Im1 z,
  -Im2 z+ Im2 z, -Im3 z+Im3 z*] by A1,Def6 .= 0 by Lm6;
  hence thesis by Def7;
end;
