
theorem
  for R being left_zeroed right_zeroed add-cancelable distributive
  well-unital associative non empty doubleLoopStr, I being add-closed
  right-ideal non empty Subset of R, J being add-closed left-ideal non empty
  Subset of R holds sqrt (I *' J) = sqrt (I /\ J)
proof
  let R be left_zeroed right_zeroed associative add-cancelable distributive
  well-unital non empty doubleLoopStr, I be add-closed right-ideal non empty
  Subset of R, J be add-closed left-idealnon empty Subset of R;
A1: now
    let u be object;
    assume u in sqrt (I *' J);
    then consider d being Element of R such that
A2: u = d and
A3: ex m being Element of NAT st d|^m in I *' J;
    consider m being Element of NAT such that
A4: d|^m in I *' J by A3;
    consider s being FinSequence of the carrier of R such that
A5: d|^m = Sum s and
A6: for i being Element of NAT st 1 <= i & i <= len s ex a,b being
    Element of R st s.i = a*b & a in I & b in J by A4;
    consider f being sequence of the carrier of R such that
A7: Sum s = f.(len s) and
A8: f.0 = 0.R and
A9: 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 f.$1 in I /\ J;
A10: now
      let j be Element of NAT;
      assume that
      0 <= j and
A11:  j < len s;
      thus P[j] implies P[j+1]
      proof
        assume f.j in I /\ J;
        then
A12:    ex g being Element of R st g = f.j & g in I & g in J;
A13:    j + 1 <= len s & 0 + 1 <= j + 1 by A11,NAT_1:13;
        then
A14:    ex a,b being Element of R st s.(j+1) = a*b & a in I & b in J by A6;
        j + 1 in Seg(len s) by A13,FINSEQ_1:1;
        then j + 1 in dom s by FINSEQ_1:def 3;
        then
A15:    s.(j+1) = s/.(j+1) by PARTFUN1:def 6;
        then
A16:    f.(j+1) = f.j + s/.(j+1) by A9,A11;
        s/.(j+1) in J by A15,A14,Def2;
        then
A17:    f.(j+1) in J by A12,A16,Def1;
        s/.(j+1) in I by A15,A14,Def3;
        then f.(j+1) in I by A12,A16,Def1;
        hence thesis by A17;
      end;
    end;
    f.0 in I & f.0 in J by A8,Th2,Th3;
    then
A18: P[0];
    for j being Element of NAT st 0 <= j&j<=len s holds P[j] from
    INT_1:sch 7(A18,A10);
    then Sum s in I /\ J by A7;
    hence u in sqrt (I /\ J) by A2,A5;
  end;
  now
    let u be object;
    assume u in sqrt (I /\ J);
    then consider d being Element of R such that
A19: u = d and
A20: ex m being Element of NAT st d|^m in I /\ J;
    consider m being Element of NAT such that
A21: d|^m in I /\ J by A20;
    set q = <*d|^m*d|^m*>;
A22: len q = 1 by FINSEQ_1:40;
A23: ex g being Element of R st d|^m = g & g in I & g in J by A21;
A24: for i being Element of NAT st 1 <= i & i <= len q ex x,y being
    Element of R st q.i = x*y & x in I & y in J
    proof
      let i be Element of NAT;
      assume
A25:  1 <= i & i <= len q;
      then i in Seg(len q) by FINSEQ_1:1;
      then i in dom q by FINSEQ_1:def 3;
      then
A26:  q.i = q/.i by PARTFUN1:def 6;
      then q/.i = q.1 by A22,A25,XXREAL_0:1
        .= d|^m*d|^m;
      hence thesis by A23,A26;
    end;
    d|^(m+m) = d|^m*d|^m by BINOM:10
      .= Sum q by BINOM:3;
    then d|^(m+m) in I *' J by A24;
    hence u in sqrt (I *' J) by A19;
  end;
  hence thesis by A1,TARSKI:2;
end;
