
theorem
  for S, T being non empty RelStr, s being Element of S, t being Element
  of T holds [:downarrow s,downarrow t:] = downarrow [s,t]
proof
  let S, T be non empty RelStr, s be Element of S, t be Element of T;
  hereby
    let x be object;
    assume x in [:downarrow s,downarrow t:];
    then consider x1, x2 being object such that
A1: x1 in downarrow s and
A2: x2 in downarrow t and
A3: x = [x1,x2] by ZFMISC_1:def 2;
    reconsider x2 as Element of T by A2;
    reconsider x1 as Element of S by A1;
    s >= x1 & t >= x2 by A1,A2,WAYBEL_0:17;
    then [s,t] >= [x1,x2] by YELLOW_3:11;
    hence x in downarrow [s,t] by A3,WAYBEL_0:17;
  end;
  let x be object;
  assume
A4: x in downarrow [s,t];
  then reconsider x9 = x as Element of [:S,T:];
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
A5: x9 = [x9`1,x9`2] by MCART_1:21;
A6: [s,t] >= x9 by A4,WAYBEL_0:17;
  then t >= x9`2 by A5,YELLOW_3:11;
  then
A7: x`2 in downarrow t by WAYBEL_0:17;
  s >= x9`1 by A5,A6,YELLOW_3:11;
  then x`1 in downarrow s by WAYBEL_0:17;
  hence thesis by A5,A7,ZFMISC_1:def 2;
end;
