
theorem Th64:
  for R being add-associative left_zeroed right_zeroed
  add-cancelable well-unital associative commutative distributive non empty
doubleLoopStr, a being Element of R holds {a}-Ideal = the set of all
a*r where r is Element
  of R
proof
  let R be add-associative left_zeroed right_zeroed add-cancelable well-unital
commutative associative distributive non empty doubleLoopStr, a be Element of
  R;
  set A = {a};
  reconsider a9 = a as Element of A by TARSKI:def 1;
  set M = the set of all Sum s where s is LinearCombination of A ;
  set N = the set of all a*r where r is Element of R ;
A1: for u being object holds u in M implies u in N
  proof
    let u be object;
    assume u in M;
    then consider s being LinearCombination of A such that
A2: u = Sum s;
    consider f being sequence of the carrier of R such that
A3: Sum s = f.(len s) and
A4: f.0 = 0.R and
A5: for j being Nat,v being Element of R st j < len s & v =
    s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12;
    defpred P[Element of NAT] means ex r being Element of R st f.($1) = a*r;
A6: now
      let j be Element of NAT;
      assume that
      0 <= j and
A7:   j < len s;
      thus P[j] implies P[j+1]
      proof
        assume ex r being Element of R st f.j = a*r;
        then consider r1 being Element of R such that
A8:     f.j = a*r1;
        0 + 1 <= j + 1 & j + 1 <= len s by A7,NAT_1:13;
        then j + 1 in Seg(len s) by FINSEQ_1:1;
        then
A9:     j + 1 in dom s by FINSEQ_1:def 3;
        then consider
        r2,r3 being Element of R, a9 being Element of A such that
A10:    s/.(j+1) = r2*a9*r3 by Def8;
        s.(j+1) = s/.(j+1) by A9,PARTFUN1:def 6;
        then f.(j+1) = f.j + s/.(j+1) by A5,A7;
        then f.(j+1) = a*r1 + r2*a*r3 by A8,A10,TARSKI:def 1
          .= a*r1 + a*(r2*r3) by GROUP_1:def 3
          .= a*(r1 + r2*r3) by VECTSP_1:def 7;
        hence thesis;
      end;
    end;
    f.0 = a*0.R by A4,BINOM:2;
    then
A11: P[0];
    for k being Element of NAT st 0 <= k & k <= len s holds P[k] from
    INT_1:sch 7 (A11,A6);
    then ex r being Element of R st Sum s = a*r by A3;
    hence thesis by A2;
  end;
A12: now
    let x be object;
    hereby
      assume x in {a}-Ideal;
      then x in {a}-RightIdeal by Th63;
      then consider f being RightLinearCombination of A such that
A13:  x = Sum f by Th62;
      f is LinearCombination of A by Th28;
      hence x in M by A13;
    end;
    assume x in M;
    then ex s being LinearCombination of A st x = Sum s;
    hence x in {a}-Ideal by Th60;
  end;
  for u being object holds u in N implies u in M
  proof
    let u be object;
    assume u in N;
    then consider r being Element of R such that
A14: u = a*r;
    set s = <* a*r *>;
    for i being set st i in dom s ex r,t being Element of R, a being
    Element of A st s/.i = r*a*t
    proof
      let i be set;
A15:  len s = 1 by FINSEQ_1:40;
      assume i in dom s;
      then i in {1} by A15,FINSEQ_1:2,def 3;
      then i = 1 by TARSKI:def 1;
      then s/.i = a*r by FINSEQ_4:16
        .= (r*a9)*1.R;
      hence thesis;
    end;
    then reconsider s as LinearCombination of A by Def8;
    Sum s = a*r by BINOM:3;
    hence thesis by A14;
  end;
  then M = N by A1,TARSKI:2;
  hence thesis by A12,TARSKI:2;
end;
