let A, x be set ; :: thesis: ( x in 3 -tuples_on A iff ex a, b, c being set st
( a in A & b in A & c in A & x = <*a,b,c*> ) )

hereby :: thesis: ( ex a, b, c being set st
( a in A & b in A & c in A & x = <*a,b,c*> ) implies x in 3 -tuples_on A )

assume x in 3 -tuples_on A ; :: thesis: ex a, b, c being set st
( a in A & b in A & c in A & x = <*a,b,c*> )
then x in { s where s is Element of A * : len s = 3 } by FINSEQ_2:def 4;
then consider s being Element of A * such that
A1: ( x = s & len s = 3 ) ;
take a = s . 1; :: thesis: ex b, c being set st
( a in A & b in A & c in A & x = <*a,b,c*> )
take b = s . 2; :: thesis: ex c being set st
( a in A & b in A & c in A & x = <*a,b,c*> )
take c = s . 3; :: thesis: ( a in A & b in A & c in A & x = <*a,b,c*> )
( x = <*a,b,c*> & rng <*a,b,c*> = {a,b,c} & a in {a,b,c} & b in {a,b,c} & c in {a,b,c} & rng s c= A ) by A1, ENUMSET1:def 1, FINSEQ_1:62, FINSEQ_2:148;
hence ( a in A & b in A & c in A & x = <*a,b,c*> ) by A1; :: thesis: verum

end;

given a, b, c being set such that A2: ( a in A & b in A & c in A & x = <*a,b,c*> ) ; :: thesis: x in 3 -tuples_on A
reconsider A = A as non empty set by A2;
reconsider a = a, b = b, c = c as Element of A by A2;
<*a,b,c*> is Element of 3 -tuples_on A by FINSEQ_2:124;
hence x in 3 -tuples_on A by A2; :: thesis: verum