
theorem
  for a being Real holds sin.(a-PI) = -sin.a & cos.(a-PI) = -cos. a
proof
  let th be Real;
  thus sin.(th- PI) = (sin.(th))* (cos.(-PI)) + (cos.(th)) *( sin.(-PI)) by
SIN_COS:74
    .= (sin.(th))* (cos.(PI)) + (cos.(th)) *(sin.(-PI)) by SIN_COS:30
    .= (sin.(th))* (cos.(PI)) + (cos.(th)) *(-(sin.(PI))) by SIN_COS:30
    .= -sin.th by SIN_COS:76;
  thus cos.(th- PI) = (cos.(th)) * (cos.(-PI)) - ( sin.(th)) *( sin.(-PI)) by
SIN_COS:74
    .= (cos.(th)) * (cos.(PI)) - ( sin.(th)) *( sin.(-PI)) by SIN_COS:30
    .= (cos.(th)) * (cos.(PI)) - ( sin.(th)) *(-(sin.(PI))) by SIN_COS:30
    .= -cos.th by SIN_COS:76;
end;
