
theorem Th4:
  for i, j being set, b1, b2 being bag of j, b19,b29 being bag of i
  st b19 = (b1|i) & b29 = (b2|i) & b1 divides b2 holds b19 divides b29
proof
  let i,j be set, b1,b2 be bag of j, b19,b29 be bag of i;
  assume that
A1: b19=(b1|i) & b29=(b2|i) and
A2: b1 divides b2;
  now
    let k be object;
A3: dom b19 = i by PARTFUN1:def 2
      .= dom b29 by PARTFUN1:def 2;
    per cases;
    suppose
A4:   not k in dom b19;
      then b19.k = {} by FUNCT_1:def 2
        .= b29.k by A3,A4,FUNCT_1:def 2;
      hence b19.k <= b29.k;
    end;
    suppose
      k in dom b19;
      then b19.k = b1.k & b29.k = b2.k by A1,A3,FUNCT_1:47;
      hence b19.k <= b29.k by A2,PRE_POLY:def 11;
    end;
  end;
  hence thesis by PRE_POLY:def 11;
end;
