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

theorem Th40:
  for x holds sinh_C/.x = sinh.x
proof
  let x;
A1: (sinh.x)*2 =2*((exp_R.x - exp_R.(-x))/2) by SIN_COS2:def 1
    .=(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;
  sinh_C/.x = sinh_C/.z .= (exp(x+0*<i>)-exp(-z))/2 by Def3
    .= (((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
    .= ((sinh.x)*2)/2 by A1;
  hence thesis;
end;
