
theorem
  for S, T being non empty RelStr, s being Element of S, t being Element
  of T holds [:uparrow s,uparrow t:] = uparrow [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 [:uparrow s,uparrow t:];
    then consider x1, x2 being object such that
A1: x1 in uparrow s and
A2: x2 in uparrow 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_0:18;
    then [s,t] <= [x1,x2] by YELLOW_3:11;
    hence x in uparrow [s,t] by A3,WAYBEL_0:18;
  end;
  let x be object;
  assume
A4: x in uparrow [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_0:18;
  then t <= x9`2 by A5,YELLOW_3:11;
  then
A7: x`2 in uparrow t by WAYBEL_0:18;
  s <= x9`1 by A5,A6,YELLOW_3:11;
  then x`1 in uparrow s by WAYBEL_0:18;
  hence thesis by A5,A7,ZFMISC_1:def 2;
end;
