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