let X1, X2, X3, X4, X5, X6, Z be set ; :: thesis: ( ( for z being set holds
( z in Z iff ex x1, x2, x3, x4, x5, x6 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & z = [x1,x2,x3,x4,x5,x6] ) ) ) implies Z = [:X1,X2,X3,X4,X5,X6:] )
assume A1: for z being set holds
( z in Z iff ex x1, x2, x3, x4, x5, x6 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & z = [x1,x2,x3,x4,x5,x6] ) ) ; :: thesis: Z = [:X1,X2,X3,X4,X5,X6:]

now

let z be set ; :: thesis: ( ( z in Z implies z in [:[:X1,X2,X3,X4,X5:],X6:] ) & ( z in [:[:X1,X2,X3,X4,X5:],X6:] implies z in Z ) )
thus ( z in Z implies z in [:[:X1,X2,X3,X4,X5:],X6:] ) :: thesis: ( z in [:[:X1,X2,X3,X4,X5:],X6:] implies z in Z )

proof

assume z in Z ; :: thesis: z in [:[:X1,X2,X3,X4,X5:],X6:]
then consider x1, x2, x3, x4, x5, x6 being set such that
A2: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 ) and
A3: ( x6 in X6 & z = [x1,x2,x3,x4,x5,x6] ) by A1;
[x1,x2,x3,x4,x5] in [:X1,X2,X3,X4,X5:] by A2, Th31;
hence z in [:[:X1,X2,X3,X4,X5:],X6:] by A3, ZFMISC_1:def 2; :: thesis: verum

end;

assume z in [:[:X1,X2,X3,X4,X5:],X6:] ; :: thesis: z in Z
then consider x12345, x6 being set such that
A4: x12345 in [:X1,X2,X3,X4,X5:] and
A5: x6 in X6 and
A6: z = [x12345,x6] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5 being set such that
A7: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 ) and
A8: x12345 = [x1,x2,x3,x4,x5] by A4, Th30;
z = [x1,x2,x3,x4,x5,x6] by A6, A8;
hence z in Z by A1, A5, A7; :: thesis: verum

end;

hence Z = [:X1,X2,X3,X4,X5,X6:] by TARSKI:1; :: thesis: verum