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,X5:] <> {} ; :: thesis: ( not [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 ) )
then A2: ( X3 <> {} & X4 <> {} ) by Th13;
A3: X5 <> {} by A1, Th13;
( X1 <> {} & X2 <> {} ) by A1, Th13;
hence ( not [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 ) ) by A2, A3, Th14; :: thesis: verum