let D, X1, X2, X3 be non empty set ; :: thesis: ( ( for a being set holds
( a in D iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 st a = [x1,x2,x3] ) ) implies D = [:X1,X2,X3:] )
assume A1: for a being set holds
( a in D iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 st a = [x1,x2,x3] ) ; :: thesis: D = [:X1,X2,X3:]

now :: thesis: for a being object holds
( ( a in D implies a in [:[:X1,X2:],X3:] ) & ( a in [:[:X1,X2:],X3:] implies a in D ) )

let a be object ; :: thesis: ( ( a in D implies a in [:[:X1,X2:],X3:] ) & ( a in [:[:X1,X2:],X3:] implies a in D ) )
thus ( a in D implies a in [:[:X1,X2:],X3:] ) :: thesis: ( a in [:[:X1,X2:],X3:] implies a in D )

proof

assume a in D ; :: thesis: a in [:[:X1,X2:],X3:]
then consider x1 being Element of X1, x2 being Element of X2, x3 being Element of X3 such that
A2: a = [x1,x2,x3] by A1;
a = [[x1,x2],x3] by A2;
hence a in [:[:X1,X2:],X3:] ; :: thesis: verum

end;

assume a in [:[:X1,X2:],X3:] ; :: thesis: a in D
then consider x12, x3 being object such that
A3: x12 in [:X1,X2:] and
A4: x3 in X3 and
A5: a = [x12,x3] by ZFMISC_1:def 2;
reconsider x3 = x3 as Element of X3 by A4;
consider x1, x2 being object such that
A6: x1 in X1 and
A7: x2 in X2 and
A8: x12 = [x1,x2] by A3, ZFMISC_1:def 2;
reconsider x2 = x2 as Element of X2 by A7;
reconsider x1 = x1 as Element of X1 by A6;
a = [x1,x2,x3] by A5, A8;
hence a in D by A1; :: thesis: verum

end;

then D = [:[:X1,X2:],X3:] by TARSKI:2;
hence D = [:X1,X2,X3:] by ZFMISC_1:def 3; :: thesis: verum