
theorem Th83:
  for R being left_zeroed left_unital non empty doubleLoopStr,
  I,J being non empty Subset of R st I,J are_co-prime holds
  I /\ J c= (I + J) *' (I /\ J)
proof
  let R be left_zeroed left_unital non empty doubleLoopStr, I,J be non empty
  Subset of R;
  assume I,J are_co-prime;
  then
A1: I + J = the carrier of R;
    let u be object;
    assume
A2: u in I /\ J;
    then reconsider u9 = u as Element of R;
    set q = <*1.R*u9*>;
A3: len q = 1 by FINSEQ_1:39;
A4: 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+J & y in I/\J
    proof
      let i be Element of NAT;
      assume
A5:   1 <= i & i <= len q;
      take 1.R,u9;
      i = 1 by A3,A5,XXREAL_0:1;
      hence thesis by A1,A2;
    end;
    Sum q = 1.R*u9 by BINOM:3
      .= u9;
    hence u in (I + J) *' (I /\ J) by A4;
end;
