reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem Th6:
  for z1,z2 being Complex holds cos_C/.(z1+z2) = (cos_C/.z1)
  *(cos_C/.z2) - (sin_C/.z1)*(sin_C/.z2)
proof
  let z1,z2 be Complex;
  reconsider z1,z2 as Element of COMPLEX by XCMPLX_0:def 2;
  set e1 = exp(<i>*z1), e2 = exp(-<i>*z1), e3 = exp(<i>*z2), e4 = exp(-<i>*z2);
  (cos_C/.z1)*(cos_C/.z2) - (sin_C/.z1)*(sin_C/.z2) =(cos_C/.z1)*(cos_C/.
  z2) - ((exp(<i>*z1) - exp(-<i>*z1))/(2*<i>))*(sin_C/.z2) by Def1
    .=(cos_C/.z1)*(cos_C/.z2) - ((exp(<i>*z1) - exp(-<i>*z1))/(2*<i>)) *((
  exp(<i>*z2) - exp(-<i>*z2))/(2*<i>)) by Def1
    .=(cos_C/.z1)*((exp(<i>*z2) + exp(-<i>*z2))/2) - ((exp(<i>*z1) - exp(-
  <i>*z1))/(2*<i>)) *((exp(<i>*z2) - exp(-<i>*z2))/(2*<i>)) by Def2
    .=((e1 + e2)/2)*((e3 + e4)/2) - ((e1 - e2)/(2*<i>))*((e3 - e4)/(2*<i>))
  by Def2
    .=( e1*e3+e1*e3 + (e2*e4+e2*e4) )/(2*2)
    .=(Re(e1*e3)+Re(e1*e3)+(Im(e1*e3)+Im(e1*e3))*<i>+(e2*e4 + e2*e4)) /(2*2)
  by COMPLEX1:81
    .=((2*Re(e1*e3)+2*Im(e1*e3)*<i>)+(e2*e4 + e2*e4)) /(2*2)
    .=((Re(2*(e1*e3))+2*Im(e1*e3)*<i>)+(e2*e4 + e2*e4)) /(2*2) by COMSEQ_3:17
    .=((Re(2*(e1*e3))+Im(2*(e1*e3))*<i>)+(e2*e4 + e2*e4)) /(2*2) by COMSEQ_3:17
    .=(2*(e1*e3)+(e2*e4 + e2*e4))/(2*2) by COMPLEX1:13
    .=(2*(e1*e3)+(Re(e2*e4)+Re(e2*e4)+(Im(e2*e4)+Im(e2*e4))*<i>)) /(2*2) by
COMPLEX1:81
    .=(2*(e1*e3)+(2*Re(e2*e4)+2*Im(e2*e4)*<i>)) /(2*2)
    .=(2*(e1*e3)+(Re(2*(e2*e4))+2*Im(e2*e4)*<i>)) /(2*2) by COMSEQ_3:17
    .=(2*(e1*e3)+(Re(2*(e2*e4))+Im(2*(e2*e4))*<i>)) /(2*2) by COMSEQ_3:17
    .=(2*(e1*e3)+2*(e2*e4))/(2*2) by COMPLEX1:13
    .=(e1*e3)/(2)+(2*(e2*e4))/(2*2)
    .=exp(<i>*z1+<i>*z2)/(2)+(e2*e4)/(2) by SIN_COS:23
    .=exp(<i>*(z1+z2))/(2)+exp(-<i>*z1+-<i>*z2)/(2) by SIN_COS:23
    .=( exp(<i>*(z1+z2))+exp(-<i>*(z1+z2)) )/(2);
  then
  cos_C/.(z1+z2) = (cos_C/.z1)*(cos_C/.z2) - (sin_C/.z1)*(sin_C/.z2) by Def2;
  hence thesis;
end;
