
theorem
  for S, T being RelStr, X being Subset of S, Y being Subset of T holds
  [:downarrow X,downarrow Y:] = downarrow [:X,Y:]
proof
  let S, T be RelStr, X be Subset of S, Y be Subset of T;
  hereby
    let x be object;
    assume x in [:downarrow X,downarrow Y:];
    then consider x1, x2 being object such that
A1: x1 in downarrow X and
A2: x2 in downarrow Y and
A3: x = [x1,x2] by ZFMISC_1:def 2;
    reconsider S9 = S, T9 = T as non empty RelStr by A1,A2;
    reconsider x1 as Element of S9 by A1;
    consider y1 being Element of S9 such that
A4: y1 >= x1 & y1 in X by A1,WAYBEL_0:def 15;
    reconsider x2 as Element of T9 by A2;
    consider y2 being Element of T9 such that
A5: y2 >= x2 & y2 in Y by A2,WAYBEL_0:def 15;
    [y1,y2] in [:X,Y:] & [y1,y2] >= [x1,x2] by A4,A5,YELLOW_3:11,ZFMISC_1:87;
    hence x in downarrow [:X,Y:] by A3,WAYBEL_0:def 15;
  end;
  let x be object;
  assume
A6: x in downarrow [:X,Y:];
A7: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
  ex a, b being object st a in the carrier of S & b in the carrier of T & x =
  [a,b] by A6,ZFMISC_1:def 2;
  then reconsider S9 = S, T9 = T as non empty RelStr;
  reconsider x9 = x as Element of [:S9,T9:] by A6;
  consider y being Element of [:S9,T9:] such that
A8: y >= x9 & y in [:X,Y:] by A6,WAYBEL_0:def 15;
  y`2 >= x9`2 & y`2 in Y by A8,MCART_1:10,YELLOW_3:12;
  then
A9: x`2 in downarrow Y by WAYBEL_0:def 15;
  y`1 >= x9`1 & y`1 in X by A8,MCART_1:10,YELLOW_3:12;
  then
A10: x`1 in downarrow X by WAYBEL_0:def 15;
  x9 = [x9`1,x9`2] by A7,MCART_1:21;
  hence thesis by A10,A9,ZFMISC_1:def 2;
end;
