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*;

theorem Th97:
  2-tuples_on D = the set of all  <*d1,d2*>
proof
  now
    let x be object;
    thus x in 2-tuples_on D implies x in the set of all  <*d1,d2*>
    proof
      assume x in 2-tuples_on D;
      then consider s such that
A1:   x = s and
A2:   len s = 2;
      2 in Seg 2;
      then
A3:   2 in dom s by A2,FINSEQ_1:def 3;
      1 in Seg 2;
      then 1 in dom s by A2,FINSEQ_1:def 3;
      then reconsider d19 = s.1, d29 = s.2 as Element of D by A3,Th9;
      s = <*d19,d29*> by A2,FINSEQ_1:44;
      hence thesis by A1;
    end;
    assume x in the set of all  <*d1,d2*>;
    then consider d1,d2 such that
A4: x = <*d1,d2*>;
    <*d1,d2*> is FinSequence of D by Th11;
    then len <*d1,d2*> = 2 & <*d1,d2*> is Element of D* by FINSEQ_1:44,def 11;
    hence x in 2-tuples_on D by A4;
  end;
  hence thesis by TARSKI:2;
end;
