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

theorem Th4:
  for z1,z2 being Complex holds sin_C/.(z1+z2) = (sin_C/.z1)
  *(cos_C/.z2) + (cos_C/.z1)*(sin_C/.z2)
proof
  let z1,z2 be Complex;
  reconsider z1,z2 as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider e1 = exp(<i>*z1), e2 = exp(-<i>*z1),
     e3 = exp(<i>*z2), e4 = exp(-<i>*z2) as Element of COMPLEX;
  (sin_C/.z1)*(cos_C/.z2) + (cos_C/.z1)*(sin_C/.z2) =((exp(<i>*z1) - exp(-
  <i>*z1))/(2*<i>))*(cos_C/.z2) + (cos_C/.z1)*(sin_C/.z2) by Def1
    .=((exp(<i>*z1) - exp(-<i>*z1))/(2*<i>))*(cos_C/.z2) + (cos_C/.z1)*((exp
  (<i>*z2) - exp(-<i>*z2))/(2*<i>)) by Def1
    .=((exp(<i>*z1) - exp(-<i>*z1))/(2*<i>))* ((exp(-<i>*z2) + exp(<i>*z2))/
  2) + (cos_C/.z1)*((exp(<i>*z2) - exp(-<i>*z2))/(2*<i>)) by Def2
    .=((exp(<i>*z1) - exp(-<i>*z1))*(exp(-<i>*z2) + exp(<i>*z2))) /(2*<i>*2)
  + ((exp(-<i>*z1) + exp(<i>*z1))/2)* ((exp(<i>*z2) - exp(-<i>*z2))/(2*<i>))
by Def2
    .=( e1*e3+e1*e3-(e1*e4+e2*e4 - (e1*e4-e2*e4)) ) /(2*<i>*2)
    .=(Re(e1*e3)+Re(e1*e3)+(Im(e1*e3)+Im(e1*e3))*<i>-(e2*e4 + e2*e4)) /(2*
  <i>*2) by COMPLEX1:81
    .=((2*Re(e1*e3)+2*Im(e1*e3)*<i>)-(e2*e4 + e2*e4)) /(2*<i>*2)
    .=((Re(2*(e1*e3))+2*Im(e1*e3)*<i>)-(e2*e4 + e2*e4)) /(2*<i>*2) by
COMSEQ_3:17
    .=((Re(2*(e1*e3))+Im(2*(e1*e3))*<i>)-(e2*e4 + e2*e4)) /(2*<i>*2) by
COMSEQ_3:17
    .=(2*(e1*e3)-(e2*e4 + e2*e4)) /(2*<i>*2) by COMPLEX1:13
    .=(2*(e1*e3)-(Re(e2*e4)+Re(e2*e4)+(Im(e2*e4)+Im(e2*e4))*<i>)) /(2*<i>*2)
  by COMPLEX1:81
    .=(2*(e1*e3)-(2*Re(e2*e4)+2*Im(e2*e4)*<i>)) /(2*<i>*2)
    .=(2*(e1*e3)-(Re(2*(e2*e4))+2*Im(e2*e4)*<i>)) /(2*<i>*2) by COMSEQ_3:17
    .=(2*(e1*e3)-(Re(2*(e2*e4))+Im(2*(e2*e4))*<i>)) /(2*<i>*2) by COMSEQ_3:17
    .=(2*(e1*e3)-2*(e2*e4)) /(2*<i>*2) by COMPLEX1:13
    .=(e1*e3)/(2*<i>) -(2*(e2*e4))/(2*<i>*2)
    .=exp(<i>*z1+<i>*z2)/(2*<i>)-(e2*e4)/(2*<i>) by SIN_COS:23
    .=exp(<i>*(z1+z2))/(2*<i>)-exp(-<i>*z1+-<i>*z2)/(2*<i>) by SIN_COS:23
    .=( exp(<i>*(z1+z2))-exp(-<i>*(z1+z2)) )/(2*<i>);
  then
  sin_C/.(z1+z2) = (sin_C/.z1)*(cos_C/.z2) + (cos_C/.z1)*(sin_C/.z2) by Def1;
  hence thesis;
end;
