let X1, X2, X3, X4, X5, X6, X7, X8, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 be set ; ( [:X1,X2,X3,X4,X5,X6,X7,X8:] <> {} & [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 ) )
assume A1:
[:X1,X2,X3,X4,X5,X6,X7,X8:] <> {}
; ( not [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 ) )
then A2:
( X3 <> {} & X4 <> {} )
by Th132;
A3:
( X7 <> {} & X8 <> {} )
by A1, Th132;
A4:
( X5 <> {} & X6 <> {} )
by A1, Th132;
( X1 <> {} & X2 <> {} )
by A1, Th132;
hence
( not [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 ) )
by A2, A4, A3, Th133; verum