
theorem
  for R being Abelian left_zeroed right_zeroed right_complementable
  add-associative associative commutative distributive well-unital non empty
doubleLoopStr, A being non empty Subset of R, a being Element of R holds a in
  A-Ideal implies {a}-Ideal c= A-Ideal
proof
  let R be left_zeroed right_zeroed right_complementable add-associative
  associative distributive well-unital commutative Abelian non empty
  doubleLoopStr, A be non empty Subset of R, a be Element of R;
  assume a in A-Ideal;
  then consider s being LinearCombination of A such that
A1: a = Sum s by Th60;
    let u be object;
    assume u in {a}-Ideal;
    then u in the set of all a*r where r is Element of R  by Th64;
    then consider r being Element of R such that
A2: u = a*r;
    set t = s*r;
A3: dom s = dom t by POLYNOM1:def 2;
    for i being set st i in dom t ex u,v being Element of R, a being
    Element of A st t/.i = u*a*v
    proof
      let i be set;
      assume
A4:   i in dom t;
      then consider u,v being Element of R, b being Element of A such that
A5:   s/.i = u*b*v by A3,Def8;
      t/.i = (u*b*v)*r by A3,A4,A5,POLYNOM1:def 2
        .= u*b*(v*r) by GROUP_1:def 3;
      hence thesis;
    end;
    then
A6: t is LinearCombination of A by Def8;
    Sum t = u by A1,A2,BINOM:5;
    hence u in A-Ideal by A6,Th60;
end;
