let X1, X2, X3, Y1, Y2, Y3 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & [:X1,X2,X3:] = [:Y1,Y2,Y3:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) )
A1: [:X1,X2,X3:] = [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
assume A2: ( X1 <> {} & X2 <> {} ) ; :: thesis: ( not X3 <> {} or not [:X1,X2,X3:] = [:Y1,Y2,Y3:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) )
assume A3: X3 <> {} ; :: thesis: ( not [:X1,X2,X3:] = [:Y1,Y2,Y3:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) )
assume [:X1,X2,X3:] = [:Y1,Y2,Y3:] ; :: thesis: ( X1 = Y1 & X2 = Y2 & X3 = Y3 )
then A4: [:[:X1,X2:],X3:] = [:[:Y1,Y2:],Y3:] by A1, ZFMISC_1:def 3;
then [:X1,X2:] = [:Y1,Y2:] by A2, A3, ZFMISC_1:110;
hence ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) by A2, A3, A4, ZFMISC_1:110; :: thesis: verum