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

theorem Th60:
  (cosh_C/.(z1+z2))*(cosh_C/.(z1-z2)) =(sinh_C/.z1)*(sinh_C/.z1) +
(cosh_C/.z2)*(cosh_C/.z2) & (cosh_C/.(z1+z2))*(cosh_C/.(z1-z2)) =(cosh_C/.z1)*(
cosh_C/.z1) + (sinh_C/.z2)*(sinh_C/.z2) & (sinh_C/.z1)*(sinh_C/.z1) + (cosh_C/.
  z2)*(cosh_C/.z2) =(cosh_C/.z1)*(cosh_C/.z1) + (sinh_C/.z2)*(sinh_C/.z2)
proof
  set s1 = sinh_C/.z1, s2 = sinh_C/.z2, c1 = cosh_C/.z1, c2 = cosh_C/.z2;
A1: (cosh_C/.(z1+z2))*(cosh_C/.(z1-z2)) = (c1*c2+s1*s2)*(cosh_C/.(z1-z2)) by
Th14
    .= (c1*c2+s1*s2)*(c1*c2-s1*s2) by Th13
    .= c1*c1*c2*c2-(s1*s1)*s2*s2;
  then
A2: (cosh_C/.(z1+z2))*(cosh_C/.(z1-z2)) = c1*c1*(c2*c2 - s2*s2)+(c1*c1-s1*s1
  )*(s2*s2)
    .= c1*c1*1+(c1*c1-s1*s1)*(s2*s2) by Th8
    .= c1*c1+1*(s2*s2) by Th8;
  (cosh_C/.(z1+z2))*(cosh_C/.(z1-z2)) = (c1*c1 - s1*s1)*(c2*c2) + s1*s1*(
  c2*c2-s2*s2) by A1
    .= 1*(c2*c2) + s1*s1*(c2*c2-s2*s2) by Th8
    .= (c2*c2) + s1*s1*1 by Th8;
  hence thesis by A2;
end;
