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

theorem Th42:
  sin_C/.(x+y*<i>) = sin.x*(cosh.y qua Real)+cos.x*sinh.y*<i>
proof
  sin_C/.(x+y*<i>) = (sin_C/.x)*(cos_C/.(y*<i>)) + (cos_C/.x)*(sin_C/.(y*
  <i>)) by Th4
    .= (sin_C/.x)*(cosh_C/.y) + (cos_C/.x)*(sin_C/.(y*<i>)) by Th16
    .= (sin_C/.x)*(cosh_C/.y) + (cos_C/.x)*((sinh_C/.y)*<i>) by Th15
    .= (sin.x)*cosh_C/.y + cos_C/.x*(<i>*sinh_C/.y) by Th38
    .= (sin.x)*cosh_C/.y + (cos.x)*(<i>*sinh_C/.y) by Th39
    .= (sin.x)*cosh_C/.y + (cos.x)*(<i>*(sinh.y)) by Th40
    .= sin.x*cosh.y+0*<i>+(0+(cos.x*sinh.y)*<i>) by Th41;
  hence thesis;
end;
