
theorem Th81:
  for R being Abelian left_zeroed right_zeroed add-cancelable
add-associative commutative associative distributive non empty doubleLoopStr,
I,J being right-ideal non empty Subset of R holds (I + J) *' (I /\ J) c= I *' J
proof
  let R be Abelian left_zeroed right_zeroed add-cancelable add-associative
  commutative associative distributive non empty doubleLoopStr, I,J be
  right-ideal non empty Subset of R;
A1: now
    let u be object;
    assume u in (I *' (I /\ J)) + (J *' (I /\ J));
    then consider a,b being Element of R such that
A2: u = a + b and
A3: a in (I *' (I /\ J)) and
A4: b in (J *' (I /\ J));
    consider s being FinSequence of the carrier of R such that
A5: b = 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 J & b in (I /\ J) by A4;
    for i being Element of NAT st 1 <= i & i <= len s ex x,y being
    Element of R st s.i = x*y & x in I & y in J
    proof
      let i be Element of NAT;
      assume 1 <= i & i <= len s;
      then
A7:   ex x,y being Element of R st s.i = x*y & x in J & y in (I /\ J) by A6;
      I /\ J c= I by XBOOLE_1:17;
      hence thesis by A7;
    end;
    then
A8: Sum s in {Sum t where t is FinSequence of the carrier of R : for i
being Element of NAT st 1 <= i & i <= len t ex a,b being Element of R st t.i =
    a*b & a in I & b in J};
    consider q being FinSequence of the carrier of R such that
A9: a = Sum q and
A10: for i being Element of NAT st 1 <= i & i <= len q ex a,b being
    Element of R st q.i = a*b & a in I & b in (I /\ J) by A3;
    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 1 <= i & i <= len q;
      then
A11:  ex x,y being Element of R st q.i = x*y & x in I & y in (I /\ J) by A10;
      I /\ J c= J by XBOOLE_1:17;
      hence thesis by A11;
    end;
    then Sum q in {Sum t where t is FinSequence of the carrier of R : for i
being Element of NAT st 1 <= i & i <= len t ex a,b being Element of R st t.i =
    a*b & a in I & b in J};
    hence u in I *' J by A2,A9,A5,A8,Def1;
  end;
  (I + J) *' (I /\ J) = (I *' (I /\ J)) + (J *' (I /\ J)) by Th80;
  hence thesis by A1;
end;
