theorem
  EXACTLY A = ATLEAST A AND ATMOST A
  proof
    thus EXACTLY A c= ATLEAST A AND ATMOST A
    proof
      let z be object; assume
A1:   z in EXACTLY A; then
      z in {x where x is Element of X: (x ref) = rng A} by Def33; then
      consider x being Element of X such that
A2:   z = x & (x ref) = rng A;
      z in {y where y is Element of X: (rng A) c= y ref} by A2; then
A3:   z in ATLEAST A by A1,Def31;
      z in {y where y is Element of X: (y ref) c= rng A} by A2; then
      z in ATMOST A by A1,Def32;
      hence thesis by A3,XBOOLE_0:def 4;
    end;
    let z be object; assume
A4: z in ATLEAST A AND ATMOST A; then
A5: z in ATLEAST A & z in ATMOST A by XBOOLE_0:def 4; then
    z in {y where y is Element of X: (y ref) c= rng A} by Def32; then
    consider a being Element of X such that
A6: z = a & (a ref) c= rng A;
    z in {y where y is Element of X: (rng A) c= y ref} by A5,Def31; then
    consider b being Element of X such that
A7: z = b & (rng A) c= b ref;
    z = a & a ref = rng A by A6,A7; then
    z in {y where y is Element of X: (y ref) = rng A};
    hence thesis by A4,Def33;
  end;
