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

theorem Th39:
  for x holds cos_C/.x = cos.x
proof
  let x;
  x in REAL by XREAL_0:def 1;
  then reconsider z = x as Element of COMPLEX by NUMBERS:11;
  cos_C/.x = cos_C/.z .= (exp(0+<i>*x) + exp(-<i>*x))/2 by Def2
    .= (((exp_R.0)*(cos.x)+(exp_R.0)*(sin.x)*<i>) + exp(-<i>*x))/2 by Th19
    .= (((exp_R 0)*(cos.x)+(exp_R.0)*(sin.x)*<i>) + exp(-<i>*x))/2 by
SIN_COS:def 23
    .= ((1*(cos.x)+1*(sin.x)*<i>) + exp(-<i>*x))/2 by SIN_COS:51,def 23
    .= ((cos.x+sin.x*<i>) + exp(0+(-x)*<i>) )/2
    .= ((cos.x+sin.x*<i>) + ((exp_R.0)*(cos.(-x))+(exp_R.0)*(sin.(-x))*<i>)
  )/2 by Th19
    .= ((cos.x+sin.x*<i>) + ((exp_R 0)*(cos.(-x))+(exp_R.0)*(sin.(-x))*<i>)
  )/2 by SIN_COS:def 23
    .= ((cos.x+sin.x*<i>) + (1*(cos.(-x))+1*(sin.(-x))*<i>) )/2 by SIN_COS:51
,def 23
    .= ((cos.x+sin.x*<i>) + (1*(cos.x)+1*(sin.(-x))*<i>) )/2 by SIN_COS:30
    .= (cos.x+(sin.x)*<i> + (cos.x+(-sin.x)*<i>))/2 by SIN_COS:30
    .= cos.x;
  hence thesis;
end;
