
theorem
  for R being right_zeroed left_add-cancelable left-distributive non
empty doubleLoopStr, I being add-closed left-ideal non empty Subset of R, J
  being Subset of R holds (I % J) *' J c= I
proof
  let R be right_zeroed left_add-cancelable left-distributive non empty
doubleLoopStr, I be add-closed left-ideal non empty Subset of R, J be Subset
  of R;
    let u be object;
    assume u in (I % J) *' J;
    then consider s being FinSequence of the carrier of R such that
A1: Sum s = u 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 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;
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
A8:     j + 1 <= len s & 0 + 1 <= j + 1 by A7,NAT_1:13;
        then consider a,b being Element of R such that
A9:     s.(j+1) = a*b and
A10:    a in I%J and
A11:    b in J by A2;
        j + 1 in Seg(len s) by A8,FINSEQ_1:1;
        then j + 1 in dom s by FINSEQ_1:def 3;
        then
A12:    s.(j+1) = s/.(j+1) by PARTFUN1:def 6;
        then
A13:    f.(j+1) = f.j + s/.(j+1) by A5,A7;
        assume
A14:    f.j in I;
        consider d being Element of R such that
A15:    a = d and
A16:    d*J c= I by A10;
        a*b in {d*i where i is Element of R : i in J} by A11,A15;
        hence thesis by A14,A12,A13,A9,A16,Def1;
      end;
    end;
A17: P[0] by A4,Th2;
    for j being Element of NAT st 0 <= j & j <= len s holds P[j] from
    INT_1:sch 7(A17,A6);
    hence u in I by A1,A3;
end;
