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

theorem
  sinh_C/.(2*z1) + sinh_C/.(2*z2) = 2*sinh_C/.(z1+z2)*cosh_C/.(z1-z2) &
  sinh_C/.(2*z1) - sinh_C/.(2*z2) = 2*sinh_C/.(z1-z2)*cosh_C/.(z1+z2)
proof
  set c1 = cosh_C/.z1, c2 = cosh_C/.z2, s1=sinh_C/.z1, s2=sinh_C/.z2;
A1: 2*sinh_C/.(z1-z2)*cosh_C/.(z1+z2) = 2*(s1*c2-c1*s2)*cosh_C/.(z1+z2) by Th12
    .= 2*(s1*c2-c1*s2)*(c1*c2+s1*s2) by Th14
    .= 2*(s1*c1*(c2*c2-s2*s2)-(s2*c2*(c1*c1)-s2*c2*(s1*s1)) )
    .= 2*(s1*c1*1-(s2*c2*(c1*c1-s1*s1))) by Th8
    .= 2*(s1*c1*1-(s2*c2*1)) by Th8
    .= 2*s1*c1-2*(s2*c2)
    .= sinh_C/.(2*z1)-2*s2*c2 by Th58;
  2*sinh_C/.(z1+z2)*cosh_C/.(z1-z2) = 2*(s1*c2+c1*s2)*cosh_C/.(z1-z2) by Th11
    .= 2*(s1*c2+c1*s2)*(c1*c2-s1*s2) by Th13
    .= 2*((c2*c2-s2*s2)*(c1*s1)+(c1*c1*(s2*c2)-s1*s1*(c2*s2)))
    .= 2*(1*(c1*s1)+(c1*c1-s1*s1)*(c2*s2)) by Th8
    .= 2*(1*(c1*s1)+1*(c2*s2)) by Th8
    .= 2*s1*c1+2*(c2*s2)
    .= sinh_C/.(2*z1) + 2*s2*c2 by Th58
    .= sinh_C/.(2*z1) + sinh_C/.(2*z2) by Th58;
  hence thesis by A1,Th58;
end;
