let A, x be set ; :: thesis: ( x in 1 -tuples_on A iff ex a being set st
( a in A & x = <*a*> ) )

hereby :: thesis: ( ex a being set st
( a in A & x = <*a*> ) implies x in 1 -tuples_on A )

assume x in 1 -tuples_on A ; :: thesis: ex a being set st
( a in A & x = <*a*> )
then x in { s where s is Element of A * : len s = 1 } by FINSEQ_2:def 4;
then consider s being Element of A * such that
A1: ( x = s & len s = 1 ) ;
take a = s . 1; :: thesis: ( a in A & x = <*a*> )
( x = <*a*> & rng <*a*> = {a} & a in {a} & rng s c= A ) by A1, FINSEQ_1:56, FINSEQ_1:57, TARSKI:def 1;
hence ( a in A & x = <*a*> ) by A1; :: thesis: verum

end;

given a being set such that A2: ( a in A & x = <*a*> ) ; :: thesis: x in 1 -tuples_on A
reconsider A = A as non empty set by A2;
reconsider a = a as Element of A by A2;
<*a*> is Element of 1 -tuples_on A by FINSEQ_2:118;
hence x in 1 -tuples_on A by A2; :: thesis: verum