let X1, X2, X3, X4, X5, X6, X7, X8, X9, Z be set ; :: thesis: ( ( for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] ) ) ) implies Z = [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] )
assume A1: for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] ) ) ; :: thesis: Z = [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]

now

let y be set ; :: thesis: ( ( y in Z implies y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] ) & ( y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] implies y in Z ) )
thus ( y in Z implies y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] ) :: thesis: ( y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] implies y in Z )

proof

assume y in Z ; :: thesis: y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:]
then consider x1, x2, x3, x4, x5, x6, x7, x8, x9 being set such that
A2: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 ) and
A3: ( x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] ) by A1;
[x1,x2,x3,x4,x5,x6,x7,x8] in [:X1,X2,X3,X4,X5,X6,X7,X8:] by A2, Th153;
hence y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] by A3, ZFMISC_1:def 2; :: thesis: verum

end;

assume y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] ; :: thesis: y in Z
then consider x12345678, x9 being set such that
A4: x12345678 in [:X1,X2,X3,X4,X5,X6,X7,X8:] and
A5: x9 in X9 and
A6: y = [x12345678,x9] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6, x7, x8 being set such that
A7: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 ) and
A8: x12345678 = [x1,x2,x3,x4,x5,x6,x7,x8] by A4, Th152;
y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] by A6, A8;
hence y in Z by A1, A5, A7; :: thesis: verum

end;

hence Z = [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] by TARSKI:1; :: thesis: verum