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 Th100:
  3-tuples_on D = the set of all  <*d1,d2,d3*>
proof
  now
    let x be object;
    thus x in 3-tuples_on D implies x in the set of all  <*d1,d2,d3*>
    proof
      assume x in 3-tuples_on D;
      then consider s such that
A1:   x = s and
A2:   len s = 3;
      2 in Seg(3);
      then
A3:   2 in dom s by A2,FINSEQ_1:def 3;
      3 in Seg(3);
      then
A4:   3 in dom s by A2,FINSEQ_1:def 3;
      1 in Seg(3);
      then 1 in dom s by A2,FINSEQ_1:def 3;
      then reconsider d19 = s.1, d29 = s.2, d39 = s.3 as Element of D by A3,A4
,Th9;
      s = <*d19,d29,d39*> by A2,FINSEQ_1:45;
      hence thesis by A1;
    end;
    assume x in the set of all  <*d1,d2,d3*>;
    then consider d1,d2,d3 such that
A5: x = <*d1,d2,d3*>;
    <*d1,d2,d3*> is FinSequence of D by Th12;
    then len <*d1,d2,d3*> = 3 & <*d1,d2,d3*> is Element of D* by FINSEQ_1:45
,def 11;
    hence x in 3-tuples_on D by A5;
  end;
  hence thesis by TARSKI:2;
end;
