
theorem
  for R being Abelian left_zeroed right_zeroed add-cancelable
  well-unital add-associative associative commutative distributive non empty
  doubleLoopStr, a,b being Element of R holds {a,b}-Ideal = {a}-Ideal + {b}
  -Ideal
proof
  let R be Abelian left_zeroed right_zeroed add-cancelable well-unital
add-associative associative commutative distributive non empty doubleLoopStr,
  a,b be Element of R;
A1: now
    let u be object;
    assume u in {a,b}-Ideal;
    then u in the set of all a*r + b*s where r,s is Element of R  by Th65;
    then consider r,s being Element of R such that
A2: u = a*r + b*s;
    b*s in the set of all b*v where v is Element of R ;
    then reconsider b9 = b*s as Element of {b}-Ideal by Th64;
    a*r in the set of all a*v where v is Element of R ;
    then reconsider a9 = a*r as Element of {a}-Ideal by Th64;
    a9 + b9 in {x + y where x,y is Element of R : x in {a}-Ideal & y in {b
    }-Ideal};
    hence u in {a}-Ideal + {b}-Ideal by A2;
  end;
  now
    let u be object;
    assume u in {a}-Ideal + {b}-Ideal;
    then consider x,y being Element of R such that
A3: u = x + y and
A4: x in {a}-Ideal and
A5: y in {b}-Ideal;
    y in the set of all b*v where v is Element of R  by A5,Th64;
    then
A6: ex s being Element of R st y = b*s;
    x in the set of all a*v where v is Element of R  by A4,Th64;
    then ex r being Element of R st x = a*r;
    then
    u in the set of all a*v + b*d where v,d is Element of R  by A3,A6;
    hence u in {a,b}-Ideal by Th65;
  end;
  hence thesis by A1,TARSKI:2;
end;
