let S, T be non empty RelStr ; :: thesis: for s being Element of S
for t being Element of T holds [:(uparrow s),(uparrow t):] = uparrow [s,t]
let s be Element of S; :: thesis: for t being Element of T holds [:(uparrow s),(uparrow t):] = uparrow [s,t]
let t be Element of T; :: thesis: [:(uparrow s),(uparrow t):] = uparrow [s,t]

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: uparrow [s,t] c= [:(uparrow s),(uparrow t):]

let x be object ; :: thesis: ( x in [:(uparrow s),(uparrow t):] implies x in uparrow [s,t] )
assume x in [:(uparrow s),(uparrow t):] ; :: thesis: x in uparrow [s,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 = x2 as Element of T by A2;
reconsider x1 = 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; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in uparrow [s,t] or x in [:(uparrow s),(uparrow t):] )
assume A4: x in uparrow [s,t] ; :: thesis: x in [:(uparrow s),(uparrow 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 x in [:(uparrow s),(uparrow t):] by A5, A7, ZFMISC_1:def 2; :: thesis: verum