let x1, x2, x3, x4, x5, x6, X1, X2, X3, X4, X5, X6 be set ; :: thesis: ( [x1,x2,x3,x4,x5,x6] in [:X1,X2,X3,X4,X5,X6:] iff ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 ) )
A1: ( [x1,x2] in [:X1,X2:] iff ( x1 in X1 & x2 in X2 ) ) by ZFMISC_1:87;
( [:X1,X2,X3,X4,X5,X6:] = [:[:X1,X2:],X3,X4,X5,X6:] & [x1,x2,x3,x4,x5,x6] = [[x1,x2],x3,x4,x5,x6] ) by Th6, Th12;
hence ( [x1,x2,x3,x4,x5,x6] in [:X1,X2,X3,X4,X5,X6:] iff ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 ) ) by A1, Th31; :: thesis: verum