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

theorem Th58:
  sinh_C/.(2*z) = 2*(sinh_C/.z)*(cosh_C/.z) & cosh_C/.(2*z) = 2*(
  cosh_C/.z)*(cosh_C/.z) - 1
proof
  set e1 = exp(z), e2 = exp(-z);
A1: 2*(cosh_C/.z)*(cosh_C/.z) - 1 = 2*((cosh_C/.z)*(cosh_C/.z)) - 1
    .= 2*((cosh_C/.(2*z) + 1)/2) - 1 by Th57
    .= cosh_C/.(2*z) + 1 - 1;
  2*(sinh_C/.z)*(cosh_C/.z) = 2*((sinh_C/.z)*(cosh_C/.z))
    .= 2*((sinh_C/.z)*((e1 + e2)/2)) by Def4
    .= 2*((e1 - e2)/2*((e1 + e2)/2)) by Def3
    .= (e1*e1 - e2*e2)/2
    .= (exp(z+z) - e2*e2)/2 by SIN_COS:23
    .= (exp(2*z) - exp(-z+-z))/2 by SIN_COS:23
    .= (exp(2*z) - exp(-(2*z)))/2
    .= sinh_C/.(2*z) by Lm1;
  hence thesis by A1;
end;
