let z, A, B, C, D, E be set ; :: thesis: ( z in [:A,B,C,D,E:] implies ( z `1_5 in A & z `2_5 in B & z `3_5 in C & z `4_5 in D & z `5_5 in E ) )
assume A1: z in [:A,B,C,D,E:] ; :: thesis: ( z `1_5 in A & z `2_5 in B & z `3_5 in C & z `4_5 in D & z `5_5 in E )
then A2: ( not A is empty & not B is empty & not C is empty & not D is empty & not E is empty ) by MCART_2:13;
then consider a being Element of A, b being Element of B, c being Element of C, d being Element of D, e being Element of E such that
A3: z = [a,b,c,d,e] by A1, MCART_2:17;
( z `1_5 = a & z `2_5 = b & z `3_5 = c & z `4_5 = d & z `5_5 = e ) by A3, Def8, Def9, Def10, Def11, Def12;
hence ( z `1_5 in A & z `2_5 in B & z `3_5 in C & z `4_5 in D & z `5_5 in E ) by A2; :: thesis: verum