let z be object ; :: thesis: for X1, X2, X3 being set st z in [:X1,X2,X3:] holds
ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] )
let X1, X2, X3 be set ; :: thesis: ( z in [:X1,X2,X3:] implies ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) )
assume z in [:X1,X2,X3:] ; :: thesis: ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] )
then z in [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
then consider x12, x3 being object such that
A1: x12 in [:X1,X2:] and
A2: x3 in X3 and
A3: z = [x12,x3] by ZFMISC_1:def 2;
consider x1, x2 being object such that
A4: ( x1 in X1 & x2 in X2 ) and
A5: x12 = [x1,x2] by A1, ZFMISC_1:def 2;
z = [x1,x2,x3] by A3, A5;
hence ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) by A2, A4; :: thesis: verum