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

theorem
  sinh_C/.z*sinh_C/.z = (cosh_C/.(2*z) - 1)/2
proof
  set e1 = exp(z), e2 = exp(-z);
  sinh_C/.z*sinh_C/.z = (exp(z) - exp(-z))/2*sinh_C/.z by Def3
    .= (e1 - e2)/2*((e1 - e2)/2) by Def3
    .= (e1*e1 + e2*e2 - 2*(e1*e2))/2/2
    .= (e1*e1 + e2*e2 - 2*1)/2/2 by Lm3
    .= (exp(z+z) + e2*e2 - 2)/2/2 by SIN_COS:23
    .= (exp(2*z) + exp(-z+-z) - 2)/2/2 by SIN_COS:23
    .= (( exp(2*z) + exp(-2*(z)) )/2 - 1)/2;
  hence thesis by Lm2;
end;
