let S, T be non empty up-complete Poset; :: thesis: for s being Element of S
for t being Element of T holds [:(waybelow s),(waybelow t):] = waybelow [s,t]
let s be Element of S; :: thesis: for t being Element of T holds [:(waybelow s),(waybelow t):] = waybelow [s,t]
let t be Element of T; :: thesis: [:(waybelow s),(waybelow t):] = waybelow [s,t]

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

let x be object ; :: thesis: ( x in [:(waybelow s),(waybelow t):] implies x in waybelow [s,t] )
assume x in [:(waybelow s),(waybelow t):] ; :: thesis: x in waybelow [s,t]
then consider x1, x2 being object such that
A1: x1 in waybelow s and
A2: x2 in waybelow 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_3:7;
then [s,t] >> [x1,x2] by Th19;
hence x in waybelow [s,t] by A3; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in waybelow [s,t] or x in [:(waybelow s),(waybelow t):] )
assume A4: x in waybelow [s,t] ; :: thesis: x in [:(waybelow s),(waybelow 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_3:7;
then t >> x9 `2 by A5, Th19;
then A7: x `2 in waybelow t ;
s >> x9 `1 by A5, A6, Th19;
then x `1 in waybelow s ;
hence x in [:(waybelow s),(waybelow t):] by A5, A7, ZFMISC_1:def 2; :: thesis: verum