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
  for A being set,a,b,c being object st <*a,b,c*> in 3-tuples_on A holds
  a in A & b in A & c in A
proof
  let A be set,a,b,c be object;
A1: <*a,b,c*>.3 = c;
  assume <*a,b,c*> in 3-tuples_on A;
  then
A2: ex a9,b9,c9 being object
   st a9 in A & b9 in A & c9 in A & <*a,b,c*> = <*a9,
  b9,c9*> by Th137;
  <*a,b,c*>.1 = a & <*a,b,c*>.2 = b;
  hence thesis by A2,A1,FINSEQ_1:45;
end;
