let X1, X2, X3, X4, X5, X6, X7, Y1, Y2, Y3, Y4, Y5, Y6, Y7 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 ) )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} ) ; :: thesis: ( not X7 <> {} or not [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 ) )
then A2: [:X1,X2,X3,X4,X5,X6:] <> {} by Th51;
assume A3: X7 <> {} ; :: thesis: ( not [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 ) )
assume A4: [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] ; :: thesis: ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 )
then [:X1,X2,X3,X4,X5,X6:] = [:Y1,Y2,Y3,Y4,Y5,Y6:] by A2, A3, ZFMISC_1:110;
hence ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 ) by A1, A2, A3, A4, Th52, ZFMISC_1:110; :: thesis: verum