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 Th135:
  for A being set, x being object holds x in 2-tuples_on A iff
  ex a,b being object st a in A & b in A & x = <*a,b*>
proof
  let A be set, x be object;
  hereby
    assume x in 2-tuples_on A;
    then x in {s where s is Element of A*: len s = 2};
    then consider s being Element of A* such that
A1: x = s and
A2: len s = 2;
    reconsider a = s.1, b = s.2 as object;
    take a,b;
A3: rng <*a,b*> = {a,b} & a in {a,b} by Lm1,TARSKI:def 2;
A4: b in {a,b} & rng s c= A by RELAT_1:def 19,TARSKI:def 2;
    x = <*a,b*> by A1,A2,FINSEQ_1:44;
    hence a in A & b in A & x = <*a,b*> by A1,A3,A4;
  end;
  given a,b being object such that
A5: a in A and
A6: b in A and
A7: x = <*a,b*>;
  reconsider A as non empty set by A5;
  reconsider a,b as Element of A by A5,A6;
  <*a,b*> is Element of 2-tuples_on A by Th99;
  hence thesis by A7;
end;
