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 Th36:
  z*' = [*Rea z, -Im1 z, -Im2 z, -Im3 z*]
proof
 <i> = [*zz,jj,zz,zz*] by Lm2,Lm3;
  then z*'= Rea z + [*(-Im1 z)*0,(-Im1 z) *1,(-Im1 z) *0,(-Im1 z) *0 *]
  + (-Im2 z)*<j> + (-Im3 z)*<k> by Def20
    .= Rea z + [*0,-Im1 z,0,0 *] +
  [*(-Im2 z)*0,(-Im2 z)*0,(-Im2 z)*1,(-Im2 z)*0*] + (-Im3 z)*<k> by Def20
    .= Rea z + [*0,-Im1 z,0,0 *] + [*0,0,(-Im2 z),0*] +
  [*(-Im3 z)*0,(-Im3 z)*0,(-Im3 z)*0,(-Im3 z)*1*] by Def20
    .= [*Rea z+0,-Im1 z,0,0 *] + [*0,0,(-Im2 z),0*]
  + [*0,0,0,(-Im3 z)*] by Def18
    .= [*Rea z+0,-Im1 z+0,0+(-Im2 z),0+0*] + [*0,0,0,(-Im3 z)*] by Def6
    .= [*Rea z+0,-Im1 z+0,-Im2 z+0,0+(-Im3 z)*] by Def6;
  hence thesis;
end;
