let S, T be antisymmetric with_infima RelStr ; :: thesis: for x, y being Element of [:S,T:] holds
( (x "/\" y) `1 = (x `1) "/\" (y `1) & (x "/\" y) `2 = (x `2) "/\" (y `2) )
let x, y be Element of [:S,T:]; :: thesis: ( (x "/\" y) `1 = (x `1) "/\" (y `1) & (x "/\" y) `2 = (x `2) "/\" (y `2) )
set a = (x "/\" y) `1 ;
set b = (x "/\" y) `2 ;
A1: the carrier of [:S,T:] = [: the carrier of S, the carrier of T:] by YELLOW_3:def 2;
then A2: x = [(x `1),(x `2)] by MCART_1:21;
A3: y = [(y `1),(y `2)] by A1, MCART_1:21;
A4: for d being Element of S st d <= x `1 & d <= y `1 holds
(x "/\" y) `1 >= d

proof

set t = (x `2) "/\" (y `2);
let d be Element of S; :: thesis: ( d <= x `1 & d <= y `1 implies (x "/\" y) `1 >= d )
assume that
A5: d <= x `1 and
A6: d <= y `1 ; :: thesis: (x "/\" y) `1 >= d
(x `2) "/\" (y `2) <= y `2 by YELLOW_0:23;
then A7: [d,((x `2) "/\" (y `2))] <= y by A3, A6, YELLOW_3:11;
(x `2) "/\" (y `2) <= x `2 by YELLOW_0:23;
then [d,((x `2) "/\" (y `2))] <= x by A2, A5, YELLOW_3:11;
then A8: x "/\" y >= [d,((x `2) "/\" (y `2))] by A7, YELLOW_0:23;
[d,((x `2) "/\" (y `2))] `1 = d ;
hence (x "/\" y) `1 >= d by A8, YELLOW_3:12; :: thesis: verum

end;

A9: for d being Element of T st d <= x `2 & d <= y `2 holds
(x "/\" y) `2 >= d

proof

set s = (x `1) "/\" (y `1);
let d be Element of T; :: thesis: ( d <= x `2 & d <= y `2 implies (x "/\" y) `2 >= d )
assume that
A10: d <= x `2 and
A11: d <= y `2 ; :: thesis: (x "/\" y) `2 >= d
(x `1) "/\" (y `1) <= y `1 by YELLOW_0:23;
then A12: [((x `1) "/\" (y `1)),d] <= y by A3, A11, YELLOW_3:11;
(x `1) "/\" (y `1) <= x `1 by YELLOW_0:23;
then [((x `1) "/\" (y `1)),d] <= x by A2, A10, YELLOW_3:11;
then A13: x "/\" y >= [((x `1) "/\" (y `1)),d] by A12, YELLOW_0:23;
[((x `1) "/\" (y `1)),d] `2 = d ;
hence (x "/\" y) `2 >= d by A13, YELLOW_3:12; :: thesis: verum

end;

x "/\" y <= y by YELLOW_0:23;
then A14: ( (x "/\" y) `1 <= y `1 & (x "/\" y) `2 <= y `2 ) by YELLOW_3:12;
x "/\" y <= x by YELLOW_0:23;
then ( (x "/\" y) `1 <= x `1 & (x "/\" y) `2 <= x `2 ) by YELLOW_3:12;
hence ( (x "/\" y) `1 = (x `1) "/\" (y `1) & (x "/\" y) `2 = (x `2) "/\" (y `2) ) by A14, A4, A9, YELLOW_0:19; :: thesis: verum