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

theorem Th59:
  (sinh_C/.z1)*(sinh_C/.z1) - (sinh_C/.z2)*(sinh_C/.z2) =(sinh_C/.
(z1+z2))*(sinh_C/.(z1-z2)) & (cosh_C/.z1)*(cosh_C/.z1) - (cosh_C/.z2)*(cosh_C/.
z2) =(sinh_C/.(z1+z2))*(sinh_C/.(z1-z2)) & (sinh_C/.z1)*(sinh_C/.z1) - (sinh_C
  /.z2)*(sinh_C/.z2) =(cosh_C/.z1)*(cosh_C/.z1) - (cosh_C/.z2)*(cosh_C/.z2)
proof
  set s1 = sinh_C/.z1, s2 = sinh_C/.z2, c1 = cosh_C/.z1, c2 = cosh_C/.z2;
A1: (sinh_C/.(z1+z2))*(sinh_C/.(z1-z2)) =(s1*c2+c1*s2)*(sinh_C/.(z1-z2)) by
Th11
    .=(s1*c2+c1*s2)*(s1*c2-c1*s2) by Th12
    .=(s1*s1)*(c2*c2)-s2*(c1*c1)*s2;
  then
A2: (sinh_C/.(z1+z2))*(sinh_C/.(z1-z2)) = (-((c1*c1)-(s1*s1)))*(c2*c2)+(c1*
  c1)*(c2*c2)-(s2*s2)*(c1*c1)
    .= (-1)*(c2*c2)+(c1*c1)*(c2*c2)-(s2*s2)*(c1*c1) by Th8
    .= (-1)*(c2*c2)+(c1*c1)*((c2*c2)-(s2*s2))
    .= (-1)*(c2*c2)+(c1*c1)*1 by Th8
    .= -(c2*c2)+c1*c1;
  (sinh_C/.(z1+z2))*(sinh_C/.(z1-z2)) =(s1*s1)*((c2*c2)-(s2*s2))+(s1*s1)*(
  s2*s2)-(s2*s2)*(c1*c1) by A1
    .=(s1*s1)*1+(s1*s1)*(s2*s2)-(s2*s2)*(c1*c1) by Th8
    .=(s1*s1)*1+(s2*s2)*-((c1*c1)-(s1*s1))
    .=(s1*s1)*1+(s2*s2)*-1 by Th8
    .=(s1*s1)*1-(s2*s2);
  hence thesis by A2;
end;
