let S, T be non empty up-complete Poset; :: thesis: for s being Element of S
for t being Element of T holds [:(compactbelow s),(compactbelow t):] = compactbelow [s,t]
let s be Element of S; :: thesis: for t being Element of T holds [:(compactbelow s),(compactbelow t):] = compactbelow [s,t]
let t be Element of T; :: thesis: [:(compactbelow s),(compactbelow t):] = compactbelow [s,t]
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by YELLOW_3:def 2;

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

let x be set ; :: thesis: ( x in [:(compactbelow s),(compactbelow t):] implies x in compactbelow [s,t] )
assume x in [:(compactbelow s),(compactbelow t):] ; :: thesis: x in compactbelow [s,t]
then consider x1, x2 being set such that
A2: ( x1 in compactbelow s & x2 in compactbelow t & x = [x1,x2] ) by ZFMISC_1:def 2;
reconsider x1 = x1 as Element of S by A2;
reconsider x2 = x2 as Element of T by A2;
( s >= x1 & t >= x2 ) by A2, WAYBEL_8:4;
then A3: [s,t] >= [x1,x2] by YELLOW_3:11;
A4: ( [x1,x2] `1 = x1 & [x1,x2] `2 = x2 ) by MCART_1:7;
( x1 is compact & x2 is compact ) by A2, WAYBEL_8:4;
then [x1,x2] is compact by A4, Th23;
hence x in compactbelow [s,t] by A2, A3; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in compactbelow [s,t] or x in [:(compactbelow s),(compactbelow t):] )
assume A5: x in compactbelow [s,t] ; :: thesis: x in [:(compactbelow s),(compactbelow t):]
then reconsider x' = x as Element of [:S,T:] ;
A6: x' = [(x' `1 ),(x' `2 )] by A1, MCART_1:23;
A7: ( [s,t] >= x' & x' is compact ) by A5, WAYBEL_8:4;
then A8: ( s >= x' `1 & t >= x' `2 ) by A6, YELLOW_3:11;
( x' `1 is compact & x' `2 is compact ) by A7, Th22;
then ( x `1 in compactbelow s & x `2 in compactbelow t ) by A8;
hence x in [:(compactbelow s),(compactbelow t):] by A6, MCART_1:11; :: thesis: verum