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

theorem Th8:
  (cosh_C/.z)*(cosh_C/.z) - (sinh_C/.z)*(sinh_C/.z) = 1
proof
  set e1 = exp(z), e2 = exp(-z);
  (cosh_C/.z)*(cosh_C/.z) - (sinh_C/.z)*(sinh_C/.z)
    =(cosh_C/.z)*(cosh_C/.z) - ((e1-e2)/2)*(sinh_C/.z) by Def3
    .=(cosh_C/.z)*(cosh_C/.z) - ((e1-e2)/2)*((e1-e2)/2) by Def3
    .=((e1+e2)/2)*(cosh_C/.z) - (((e1-e2)*(e1-e2))/(2*2)) by Def4
    .=((e1+e2)/2)*((e1+e2)/2) - (((e1-e2)*(e1-e2))/(2*2)) by Def4
    .=(e1*e2 + e1*e2 + (e1*e2 + e1*e2))/(2*2)
    .=(1 + e1*e2 + (e1*e2 + e1*e2))/(2*2) by Lm3
    .=(1 + 1 + (e1*e2 + e1*e2))/(2*2) by Lm3
    .=(1 + 1 + (1 + e1*e2))/(2*2) by Lm3
    .=(2+2)/(2*2) by Lm3
    .= 1;
  hence thesis;
end;
