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

theorem Th38:
  for x holds sin_C/.x = sin.x
proof
  let x;
  x in REAL by XREAL_0:def 1;
  then reconsider z = x as Element of COMPLEX by NUMBERS:11;
  sin_C/.x = sin_C/.z .= (exp(-0+x*<i>) - exp(-<i>*x))/(2*<i>) by Def1
    .= ((exp_R.0)*(cos.x)+(exp_R.0)*(sin.x)*<i> - exp(-(x*<i>)))/(2*<i>) by
Th19
    .= ((exp_R 0)*(cos.x)+(exp_R.0)*(sin.x)*<i> - exp(-(x*<i>)))/(2*<i>) by
SIN_COS:def 23
    .= ((exp_R 0)*(cos.x)+(exp_R 0)*(sin.x)*<i> - exp(-(x*<i>)))/(2*<i>) by
SIN_COS:def 23
    .= (cos.x+sin.x*<i> - exp(0+(-x)*<i>))/(2*<i>) by SIN_COS:51
    .= (cos.x+sin.x*<i> - ((exp_R.0)*(cos.-x)+(exp_R.0)*(sin.-x)*<i>))/(2*
  <i>) by Th19
    .= (cos.x+sin.x*<i> - ((exp_R 0)*(cos.-x)+(exp_R.0)*(sin.-x)*<i>))/(2*
  <i>) by SIN_COS:def 23
    .= (cos.x+sin.x*<i> - (1*(cos.-x)+1*(sin.-x)*<i>))/(2*<i>) by SIN_COS:51
,def 23
    .= (cos.x+sin.x*<i> - (cos.-x+(-(sin.x))*<i>))/(2*<i>) by SIN_COS:30
    .= (cos.x+sin.x*<i> - (cos.x+(-(sin.x))*<i>))/(2*<i>) by SIN_COS:30
    .= sin.x;
  hence thesis;
end;
