set IT = [:X,Y:];
let x, y be Element of [:X,Y:]; :: according to LATTICE3:def 11 :: thesis: ex b₁ being Element of the carrier of [:X,Y:] st
( b₁ <= x & b₁ <= y & ( for b₂ being Element of the carrier of [:X,Y:] holds
( not b₂ <= x or not b₂ <= y or b₂ <= b₁ ) ) )
consider zx being Element of X such that
A1: ( x `1 >= zx & y `1 >= zx ) and
A2: for z9 being Element of X st x `1 >= z9 & y `1 >= z9 holds
zx >= z9 by LATTICE3:def 11;
consider zy being Element of Y such that
A3: ( x `2 >= zy & y `2 >= zy ) and
A4: for z9 being Element of Y st x `2 >= z9 & y `2 >= z9 holds
zy >= z9 by LATTICE3:def 11;
A5: the carrier of [:X,Y:] = [: the carrier of X, the carrier of Y:] by Def2;
then A6: ( ex a, b being object st
( a in the carrier of X & b in the carrier of Y & x = [a,b] ) & ex c, d being object st
( c in the carrier of X & d in the carrier of Y & y = [c,d] ) ) by ZFMISC_1:def 2;
take z = [zx,zy]; :: thesis: ( z <= x & z <= y & ( for b₁ being Element of the carrier of [:X,Y:] holds
( not b₁ <= x or not b₁ <= y or b₁ <= z ) ) )
( [(x `1),(x `2)] >= [zx,zy] & [(y `1),(y `2)] >= [zx,zy] ) by A1, A3, Th11;
hence ( x >= z & y >= z ) by A6; :: thesis: for b₁ being Element of the carrier of [:X,Y:] holds
( not b₁ <= x or not b₁ <= y or b₁ <= z )
let z9 be Element of [:X,Y:]; :: thesis: ( not z9 <= x or not z9 <= y or z9 <= z )
A7: ex a, b being object st
( a in the carrier of X & b in the carrier of Y & z9 = [a,b] ) by A5, ZFMISC_1:def 2;
assume A8: ( x >= z9 & y >= z9 ) ; :: thesis: z9 <= z
then ( x `2 >= z9 `2 & y `2 >= z9 `2 ) by Th12;
then A9: zy >= z9 `2 by A4;
( x `1 >= z9 `1 & y `1 >= z9 `1 ) by A8, Th12;
then zx >= z9 `1 by A2;
then [zx,zy] >= [(z9 `1),(z9 `2)] by A9, Th11;
hence z9 <= z by A7; :: thesis: verum