
theorem Th63:
  for R being add-associative left_zeroed right_zeroed
  add-cancelable well-unital associative commutative distributive non empty
  doubleLoopStr, F being non empty Subset of R holds F-Ideal = F-LeftIdeal & F
  -Ideal = F-RightIdeal
proof
  let R be add-associative left_zeroed right_zeroed add-cancelable well-unital
associative commutative distributive non empty doubleLoopStr, F be non empty
  Subset of R;
  now
    let x be object;
    hereby
      assume x in F-Ideal;
      then consider lc being LinearCombination of F such that
A1:   x = Sum lc by Th60;
      lc is LeftLinearCombination of F by Th31;
      hence x in F-LeftIdeal by A1,Th61;
    end;
    assume x in F-LeftIdeal;
    then consider lc being LeftLinearCombination of F such that
A2: x = Sum lc by Th61;
    lc is LinearCombination of F by Th25;
    hence x in F-Ideal by A2,Th60;
  end;
  hence F-Ideal = F-LeftIdeal by TARSKI:2;
  now
    let x be object;
    hereby
      assume x in F-Ideal;
      then consider lc being LinearCombination of F such that
A3:   x = Sum lc by Th60;
      lc is RightLinearCombination of F by Th31;
      hence x in F-RightIdeal by A3,Th62;
    end;
    assume x in F-RightIdeal;
    then consider lc being RightLinearCombination of F such that
A4: x = Sum lc by Th62;
    lc is LinearCombination of F by Th28;
    hence x in F-Ideal by A4,Th60;
  end;
  hence thesis by TARSKI:2;
end;
