let L be with_infima Poset; :: thesis: for X, Y being Subset of
for X', Y' being Subset of st X = X' & Y = Y' holds
X "/\" Y = X' "\/" Y'
let X, Y be Subset of ; :: thesis: for X', Y' being Subset of st X = X' & Y = Y' holds
X "/\" Y = X' "\/" Y'
let X', Y' be Subset of ; :: thesis: ( X = X' & Y = Y' implies X "/\" Y = X' "\/" Y' )
assume A1: ( X = X' & Y = Y' ) ; :: thesis: X "/\" Y = X' "\/" Y'
thus X "/\" Y c= X' "\/" Y' :: according to XBOOLE_0:def 10 :: thesis: X' "\/" Y' c= X "/\" Y

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in X "/\" Y or a in X' "\/" Y' )
assume a in X "/\" Y ; :: thesis: a in X' "\/" Y'
then a in { (p "/\" q) where p, q is Element of : ( p in X & q in Y ) } by YELLOW_4:def 4;
then consider x, y being Element of such that
A2: a = x "/\" y and
A3: ( x in X & y in Y ) ;
A4: a = (x ~ ) "\/" (y ~ ) by A2, YELLOW_7:21;
( x ~ in X' & y ~ in Y' ) by A1, A3, LATTICE3:def 6;
then a in { (p "\/" q) where p, q is Element of : ( p in X' & q in Y' ) } by A4;
hence a in X' "\/" Y' by YELLOW_4:def 3; :: thesis: verum

end;

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in X' "\/" Y' or a in X "/\" Y )
assume a in X' "\/" Y' ; :: thesis: a in X "/\" Y
then a in { (p "\/" q) where p, q is Element of : ( p in X' & q in Y' ) } by YELLOW_4:def 3;
then consider x, y being Element of such that
A5: a = x "\/" y and
A6: ( x in X' & y in Y' ) ;
A7: a = (~ x) "/\" (~ y) by A5, YELLOW_7:22;
( ~ x in X & ~ y in Y ) by A1, A6, LATTICE3:def 7;
then a in { (p "/\" q) where p, q is Element of : ( p in X & q in Y ) } by A7;
hence a in X "/\" Y by YELLOW_4:def 4; :: thesis: verum