reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;
reserve m,n for Nat,
  s,w for FinSequence of NAT;

theorem Th137:
  for A being set, x being object holds x in 3-tuples_on A iff
  ex a,b,c being object st a in A & b in A & c in A & x = <*a,b,c*>
proof
  let A be set, x be object;
  hereby
    assume x in 3-tuples_on A;
    then x in {s where s is Element of A*: len s = 3};
    then consider s being Element of A* such that
A1: x = s and
A2: len s = 3;
    reconsider a = s.1, b = s.2, c = s.3 as object;
    take a,b,c;
A3: rng <*a,b,c*> = {a,b,c} & a in {a,b,c} by Lm2,ENUMSET1:def 1;
A4: rng s c= A by RELAT_1:def 19;
A5: b in {a,b,c} & c in {a,b,c} by ENUMSET1:def 1;
    x = <*a,b,c*> by A1,A2,FINSEQ_1:45;
    hence a in A & b in A & c in A & x = <*a,b,c*> by A1,A3,A5,A4;
  end;
  given a,b,c being object such that
A6: a in A and
A7: b in A & c in A and
A8: x = <*a,b,c*>;
  reconsider A as non empty set by A6;
  reconsider a,b,c as Element of A by A6,A7;
  <*a,b,c*> is Element of 3-tuples_on A by Th102;
  hence thesis by A8;
end;
