let X1, X2, X3, X4, X5, X6, X7, Y1, Y2, Y3, Y4, Y5, Y6, Y7 be set ; :: thesis: ( [:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] implies ( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets Y6 & X7 meets Y7 ) )
A1: ( [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] = [:X1,X2,X3,X4,X5,X6,X7:] & [:[:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] ) by MC347;
assume [:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] ; :: thesis: ( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets Y6 & X7 meets Y7 )
then A2: ( [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:] meets [:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:] & X7 meets Y7 ) by A1, ZFMISC_1:127;
then ( [:[:[:[:X1,X2:],X3:],X4:],X5:] meets [:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:] & X6 meets Y6 ) by ZFMISC_1:127;
then A3: ( [:[:[:X1,X2:],X3:],X4:] meets [:[:[:Y1,Y2:],Y3:],Y4:] & X5 meets Y5 ) by ZFMISC_1:127;
then ( [:[:X1,X2:],X3:] meets [:[:Y1,Y2:],Y3:] & X4 meets Y4 ) by ZFMISC_1:127;
then ( [:X1,X2:] meets [:Y1,Y2:] & X3 meets Y3 ) by ZFMISC_1:127;
hence ( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets Y6 & X7 meets Y7 ) by A2, A3, ZFMISC_1:127; :: thesis: verum