
theorem Th70:
  for R being right_zeroed left_add-cancelable left-distributive
  non empty doubleLoopStr, I being non empty Subset of R holds 0.R*I = {0.R}
proof
  let R be right_zeroed left_add-cancelable left-distributive non empty
  doubleLoopStr, I be non empty Subset of R;
A1: now
    set j = the Element of I;
    let u be object;
    assume u in {0.R};
    then
A2: u = 0.R by TARSKI:def 1;
    0.R*j = 0.R by BINOM:1;
    hence u in 0.R*I by A2;
  end;
  now
    let u be object;
    assume u in 0.R*I;
    then ex i being Element of R st u = 0.R*i & i in I;
    then u = 0.R by BINOM:1;
    hence u in {0.R} by TARSKI:def 1;
  end;
  hence thesis by A1,TARSKI:2;
end;
