theorem Th43:
  sin_C/.(x+(-y)*<i>) = sin.x*cosh.y+(-cos.x*sinh.y)*<i>
proof
  sin_C/.(x+(-y)*<i>) = sin.x*cosh.(-y)+cos.x*sinh.(-y)*<i> by Th42
    .= sin.x*cosh.y+cos.x*sinh.(-y)*<i> by SIN_COS2:19
    .= sin.x*cosh.y+cos.x*(-sinh.y)*<i> by SIN_COS2:19;
  hence thesis;
end;
