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 Th133:
  for A being set, x being object holds
  x in 1-tuples_on A iff ex a being set st a in A & x = <*a*>
proof
  let A be set, x be object;
  hereby
    assume x in 1-tuples_on A;
    then x in {s where s is Element of A*: len s = 1};
    then consider s being Element of A* such that
A1: x = s and
A2: len s = 1;
    take a = s.1;
A3: rng <*a*> = {a} & a in {a} by FINSEQ_1:39,TARSKI:def 1;
A4: rng s c= A by RELAT_1:def 19;
    x = <*a*> by A1,A2,FINSEQ_1:40;
    hence a in A & x = <*a*> by A1,A3,A4;
  end;
  given a being set such that
A5: a in A and
A6: x = <*a*>;
  reconsider A as non empty set by A5;
  reconsider a as Element of A by A5;
  <*a*> is Element of 1-tuples_on A by Th96;
  hence thesis by A6;
end;
