
theorem
  for R being right_zeroed left_add-cancelable left-distributive non
  empty doubleLoopStr, I,J being add-closed left-ideal non empty Subset of R
  holds (I + J) *' (I /\ J) c= I /\ J
proof
  let R be right_zeroed left_add-cancelable left-distributive non empty
  doubleLoopStr, I,J be add-closed left-ideal non empty Subset of R;
    let u be object;
    assume u in (I + J) *' (I /\ J);
    then consider s being FinSequence of the carrier of R such that
A1: u = Sum s and
A2: 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+J & b in I/\J;
    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 f.$1 in I/\J;
A6: now
      let n be Element of NAT;
      assume that
      0 <= n and
A7:   n < len s;
      thus P[n] implies P[n+1]
      proof
A8:     0 + 1 <= n + 1 & n + 1 <= len s by A7,NAT_1:13;
        then n + 1 in Seg(len s) by FINSEQ_1:1;
        then n + 1 in dom s by FINSEQ_1:def 3;
        then
A9:     s.(n+1) = s/.(n+1) by PARTFUN1:def 6;
        assume
A10:    f.n in I /\ J;
        ex x,y being Element of R st s.(n+1) = x*y & x in I+J & y in I/\J
        by A2,A8;
        then s/.(n+1) in I /\ J by A9,Def2;
        then f.n + s/.(n+1) in I /\ J by A10,Def1;
        hence thesis by A5,A7,A9;
      end;
    end;
A11: P[0] by A4,Th2;
    for n being Element of NAT st 0 <= n &n <= len s holds P[n] from
    INT_1:sch 7 (A11,A6);
    hence u in I /\ J by A1,A3;
end;
