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

theorem Th41:
  for x holds cosh_C/.x = cosh.x
proof
  let x;
A1: (cosh.x)*2 =2*((exp_R.x + exp_R.(-x))/2) by SIN_COS2:def 3
    .=(exp_R.x + exp_R.(-x))/(2/2);
  x in REAL by XREAL_0:def 1;
  then reconsider z = x as Element of COMPLEX by NUMBERS:11;
  cosh_C/.x = cosh_C/.z .= (exp(x+0*<i>) + exp(-z))/2 by Def4
    .= (( (exp_R.x)*1+(exp_R.x)*0*<i>) + exp(-z))/2 by Th19,SIN_COS:30
    .= (exp_R.x + exp(-x+0*<i>))/2
    .= (exp_R.x + ((exp_R.(-x))*1+(exp_R.(-x))*0*<i>))/2 by Th19,SIN_COS:30
    .= ((cosh.x)*2)/2 by A1;
  hence thesis;
end;
