let X1, X2, X3 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} implies for x being Element of [:X1,X2,X3:] holds x = [(x `1 ),(x `2 ),(x `3 )] )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3:] holds x = [(x `1 ),(x `2 ),(x `3 )]
let x be Element of [:X1,X2,X3:]; :: thesis: x = [(x `1 ),(x `2 ),(x `3 )]
consider xx1 being Element of X1, xx2 being Element of X2, xx3 being Element of X3 such that
A2: x = [xx1,xx2,xx3] by A1, Lm2;
thus x = [(x `1 ),xx2,xx3] by A1, A2, Def5
.= [(x `1 ),(x `2 ),xx3] by A1, A2, Def6
.= [(x `1 ),(x `2 ),(x `3 )] by A1, A2, Def7 ; :: thesis: verum