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

theorem Th44:
  cos_C/.(x+y*<i>) = cos.x*cosh.y+(-sin.x*sinh.y)*<i>
proof
  cos_C/.(x+y*<i>) = cos_C/.x*cos_C/.(<i>*y)-sin_C/.x*sin_C/.(<i>*y) by Th6
    .= cos_C/.x*cos_C/.(<i>*y)-sin_C/.x*(sinh_C/.y*<i>) by Th15
    .= cos_C/.x*cosh_C/.y-sin_C/.x*(sinh_C/.y*<i>) by Th16
    .= cos_C/.x*cosh_C/.y-(sin.x)*(sinh_C/.y*<i>) by Th38
    .= (cos.x)*cosh_C/.y-(sin.x)*(sinh_C/.y*<i>) by Th39
    .= (cos.x)*cosh_C/.y-(sin.x)*((sinh.y)*<i>) by Th40
    .= (cos.x)*(cosh.y)-(sin.x)*(sinh.y)*<i> by Th41
    .= cos.x*cosh.y+(-sin.x*sinh.y)*<i>;
  hence thesis;
end;
