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

theorem Th57:
  cosh_C/.z*cosh_C/.z = (cosh_C/.(2*z) + 1)/2
proof
  set e1 = exp(z), e2 = exp(-z);
  cosh_C/.z*cosh_C/.z = (exp(z) + exp(-z))/2*cosh_C/.z by Def4
    .= (e1 + e2)/2*((e1 + e2)/2) by Def4
    .= (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;
