let X1, X2, X3, X4, Y1, Y2, Y3, Y4 be set ; ( [:X1,X2,X3,X4:] <> {} & [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
assume A1:
[:X1,X2,X3,X4:] <> {}
; ( not [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
then A2:
( X3 <> {} & X4 <> {} )
by Th39;
( X1 <> {} & X2 <> {} )
by A1, Th39;
hence
( not [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
by A2, Th40; verum