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

theorem Th13:
  cosh_C/.(z1-z2) = (cosh_C/.z1)*(cosh_C/.z2) - (sinh_C/.z1)*( sinh_C/.z2)
proof
  set e1=exp(z1), e2=exp(-z1), e3=exp(z2), e4=exp(-z2);
  (cosh_C/.z1)*(cosh_C/.z2) - (sinh_C/.z1)*(sinh_C/.z2) =((e1+e2)/2)*(
  cosh_C/.z2) - (sinh_C/.z1)*(sinh_C/.z2) by Def4
    .=((e1+e2)/2)*((e3+e4)/2) - (sinh_C/.z1)*(sinh_C/.z2) by Def4
    .=((e1+e2)*(e3+e4))/(2*2) - ((e1-e2)/2)*(sinh_C/.z2) by Def3
    .=((e1+e2)*(e3+e4))/(2*2)-((e1-e2)/2)*((e3-e4)/2) by Def3
    .=(e2*e3+e2*e3+(e1*e4+e2*e4+(e1*e4-e2*e4)))/(2*2)
    .=(Re(e2*e3)+Re(e2*e3)+(Im(e2*e3)+Im(e2*e3))*<i>+(e1*e4+e1*e4)) /(2*2)
  by COMPLEX1:81
    .=((2*Re(e2*e3)+2*Im(e2*e3)*<i>)+(e1*e4+e1*e4))/(2*2)
    .=((Re(2*(e2*e3))+2*Im(e2*e3)*<i>)+(e1*e4+e1*e4)) /(2*2) by COMSEQ_3:17
    .=((Re(2*(e2*e3))+Im(2*(e2*e3))*<i>)+(e1*e4+e1*e4)) /(2*2) by COMSEQ_3:17
    .=(2*(e2*e3)+(e1*e4+e1*e4))/(2*2) by COMPLEX1:13
    .=(2*(e2*e3)+(Re(e1*e4)+Re(e1*e4)+(Im(e1*e4)+Im(e1*e4))*<i>)) /(2*2) by
COMPLEX1:81
    .=(2*(e2*e3)+(2*Re(e1*e4)+2*Im(e1*e4)*<i>)) /(2*2)
    .=(2*(e2*e3)+(Re(2*(e1*e4))+2*Im(e1*e4)*<i>)) /(2*2) by COMSEQ_3:17
    .=(2*(e2*e3)+(Re(2*(e1*e4))+Im(2*(e1*e4))*<i>)) /(2*2) by COMSEQ_3:17
    .=(2*(e2*e3)+2*(e1*e4))/(2*2) by COMPLEX1:13
    .=(e1*e4)/(2)+(2*(e2*e3))/(2*2)
    .=exp(z1+-z2)/2+(e2*e3)/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 Def4;
end;
