reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th14:
  for a1,a2,a3,a4 being object st a1 <> a2 & a1 <> a3 & a1 <> a4 &
  a2 <> a3 & a2 <> a4 & a3 <> a4
  holds <*a1,a2,a3,a4*> is one-to-one
  proof
    let a1,a2,a3,a4 be object;
    assume A1: a1 <> a2;
    assume A2: a1 <> a3;
    assume A3: a1 <> a4;
    assume A4: a2 <> a3;
    assume A5: a2 <> a4;
    assume A6: a3 <> a4;
A7: dom <*a1,a2,a3,a4*> = Seg 4 by FINSEQ_1:89;
    let x,y be object; assume x in dom <*a1,a2,a3,a4*>;
    then
A8: x = 1 or x = 2 or x = 3 or x = 4 by A7,ENUMSET1:def 2,FINSEQ_3:2;
    assume
A9: y in dom <*a1,a2,a3,a4*>;
    <*a1,a2,a3,a4*>.1 = a1 & <*a1,a2,a3,a4*>.2 = a2 & <*a1,a2,a3,a4*>.3 = a3 &
    <*a1,a2,a3,a4*>.4 = a4;
    hence thesis by A1,A2,A3,A4,A5,A6,A8,A7,A9,ENUMSET1:def 2,FINSEQ_3:2;
  end;
