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