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

theorem Th11:
  sinh_C/.(z1+z2) = (sinh_C/.z1)*(cosh_C/.z2) + (cosh_C/.z1)*( sinh_C/.z2)
proof
  set e1=exp(z1), e2=exp(-z1), e3=exp(z2), e4=exp(-z2);
  (sinh_C/.z1)*(cosh_C/.z2) + (cosh_C/.z1)*(sinh_C/.z2) =((e1-e2)/2)*(
  cosh_C/.z2) + (cosh_C/.z1)*(sinh_C/.z2) by Def3
    .=((e1-e2)/2)*(cosh_C/.z2) + (cosh_C/.z1)*((e3-e4)/2) by Def3
    .=((e1-e2)/2)*(cosh_C/.z2) + ((e1+e2)/2)*((e3-e4)/2) by Def4
    .=((e1-e2)/2)*((e3+e4)/2) + ((e1+e2)/2)*((e3-e4)/2) by Def4
    .=(e1*e3+e1*e3-(e2*e4+e2*e4))/4
    .=(Re(e1*e3)+Re(e1*e3)+(Im(e1*e3)+Im(e1*e3))*<i>-(e2*e4 + e2*e4)) /4 by
COMPLEX1:81
    .=((2*Re(e1*e3)+2*Im(e1*e3)*<i>)-(e2*e4 + e2*e4)) /4
    .=((Re(2*(e1*e3))+2*Im(e1*e3)*<i>)-(e2*e4 + e2*e4)) /4 by COMSEQ_3:17
    .=((Re(2*(e1*e3))+Im(2*(e1*e3))*<i>)-(e2*e4 + e2*e4)) /4 by COMSEQ_3:17
    .=(2*(e1*e3)-(e2*e4 + e2*e4))/4 by COMPLEX1:13
    .=(2*(e1*e3)-(Re(e2*e4)+Re(e2*e4)+(Im(e2*e4)+Im(e2*e4))*<i>)) /4 by
COMPLEX1:81
    .=(2*(e1*e3)-(2*Re(e2*e4)+2*Im(e2*e4)*<i>)) /4
    .=(2*(e1*e3)-(Re(2*(e2*e4))+2*Im(e2*e4)*<i>)) /4 by COMSEQ_3:17
    .=(2*(e1*e3)-(Re(2*(e2*e4))+Im(2*(e2*e4))*<i>)) /4 by COMSEQ_3:17
    .=(2*(e1*e3)-2*(e2*e4))/4 by COMPLEX1:13
    .=(e1*e3)/(2)-(2*(e2*e4))/(2*2)
    .=exp(z1+z2)/2-(e2*e4)/(2) by SIN_COS:23
    .=exp(z1+z2)/2-exp(-z1+-z2)/2 by SIN_COS:23
    .=(exp(z1+z2)-exp(-(z1+z2)))/2;
  hence thesis by Def3;
end;
