let X1, X2, X3, X4, Y1, Y2, Y3, Y4 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
A1: [:X1,X2,X3,X4:] = [:[:X1,X2,X3:],X4:] by ZFMISC_1:def 4;
assume A2: ( X1 <> {} & X2 <> {} & X3 <> {} ) ; :: thesis: ( not X4 <> {} or not [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
assume A3: X4 <> {} ; :: thesis: ( not [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
assume [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] ; :: thesis: ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 )
then A4: [:[:X1,X2,X3:],X4:] = [:[:Y1,Y2,Y3:],Y4:] by A1, ZFMISC_1:def 4;
[:X1,X2,X3:] = [:Y1,Y2,Y3:] by A3, A4, A2, ZFMISC_1:110;
hence ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) by A2, A3, A4, Th22, ZFMISC_1:110; :: thesis: verum