set IT = [:X,Y:];
let x, y be Element of [:X,Y:]; :: according to LATTICE3:def 10 :: thesis: ex b₁ being Element of the carrier of [:X,Y:] st
( x <= b₁ & y <= b₁ & ( for b₂ being Element of the carrier of [:X,Y:] holds
( not x <= b₂ or not y <= b₂ or b₁ <= b₂ ) ) )
consider zx being Element of X such that
A1: ( x `1 <= zx & y `1 <= zx ) and
A2: for z' being Element of X st x `1 <= z' & y `1 <= z' holds
zx <= z' by LATTICE3:def 10;
consider zy being Element of Y such that
A3: ( x `2 <= zy & y `2 <= zy ) and
A4: for z' being Element of Y st x `2 <= z' & y `2 <= z' holds
zy <= z' by LATTICE3:def 10;
A5: the carrier of [:X,Y:] = [:the carrier of X,the carrier of Y:] by Def2;
then consider a, b being set such that
A6: ( a in the carrier of X & b in the carrier of Y & x = [a,b] ) by ZFMISC_1:def 2;
consider c, d being set such that
A7: ( c in the carrier of X & d in the carrier of Y & y = [c,d] ) by A5, ZFMISC_1:def 2;
take z = [zx,zy]; :: thesis: ( x <= z & y <= z & ( for b₁ being Element of the carrier of [:X,Y:] holds
( not x <= b₁ or not y <= b₁ or z <= b₁ ) ) )
( [(x `1 ),(x `2 )] <= [zx,zy] & [(y `1 ),(y `2 )] <= [zx,zy] ) by A1, A3, Th11;
hence ( x <= z & y <= z ) by A6, A7, MCART_1:8; :: thesis: for b₁ being Element of the carrier of [:X,Y:] holds
( not x <= b₁ or not y <= b₁ or z <= b₁ )
let z' be Element of [:X,Y:]; :: thesis: ( not x <= z' or not y <= z' or z <= z' )
consider a, b being set such that
A8: ( a in the carrier of X & b in the carrier of Y & z' = [a,b] ) by A5, ZFMISC_1:def 2;
assume ( x <= z' & y <= z' ) ; :: thesis: z <= z'
then ( x `1 <= z' `1 & x `2 <= z' `2 & y `1 <= z' `1 & y `2 <= z' `2 ) by Th12;
then ( zx <= z' `1 & zy <= z' `2 ) by A2, A4;
then [zx,zy] <= [(z' `1 ),(z' `2 )] by Th11;
hence z <= z' by A8, MCART_1:8; :: thesis: verum