let y, X1, X2, X3, X4, X5, X6, X7 be set ; :: thesis: ( y in [:X1,X2,X3,X4,X5,X6,X7:] implies ex x1, x2, x3, x4, x5, x6, x7 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] ) )
assume y in [:X1,X2,X3,X4,X5,X6,X7:] ; :: thesis: ex x1, x2, x3, x4, x5, x6, x7 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] )
then consider x123456, x7 being set such that
A1: x123456 in [:X1,X2,X3,X4,X5,X6:] and
A2: x7 in X7 and
A3: y = [x123456,x7] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6 being set such that
A4: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 ) and
A5: x123456 = [x1,x2,x3,x4,x5,x6] by A1, Th69;
y = [x1,x2,x3,x4,x5,x6,x7] by A3, A5;
hence ex x1, x2, x3, x4, x5, x6, x7 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] ) by A2, A4; :: thesis: verum