let X1, X2, X3, X4, X5, Y1, Y2, Y3, Y4, Y5 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 ) )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) ; :: thesis: ( not X5 <> {} or not [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 ) )
then A2: [:X1,X2,X3,X4:] <> {} by MCART_1:55;
assume A3: X5 <> {} ; :: thesis: ( not [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 ) )
assume A4: [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] ; :: thesis: ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 )
then [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] by A2, A3, ZFMISC_1:134;
hence ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 ) by A1, A2, A3, A4, MCART_1:56, ZFMISC_1:134; :: thesis: verum