let X1, X2, X3, Y1, Y2, Y3 be set ; ( [:X1,X2,X3:] meets [:Y1,Y2,Y3:] implies ( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 ) )
assume A1:
[:X1,X2,X3:] meets [:Y1,Y2,Y3:]
; ( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 )
A2:
( [:[:X1,X2:],X3:] = [:X1,X2,X3:] & [:[:Y1,Y2:],Y3:] = [:Y1,Y2,Y3:] )
by ZFMISC_1:def 3;
then
[:X1,X2:] meets [:Y1,Y2:]
by A1, ZFMISC_1:104;
hence
( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 )
by A2, A1, ZFMISC_1:104; verum