reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th5:
  for S being non void Signature
  for X being non-empty ManySortedSet of the carrier of S
  for t being Term of S, X
  holds t is non pair
proof
  let S be non void Signature;
  let X be non-empty ManySortedSet of the carrier of S;
  let t be Term of S, X;
  given x,y being object such that
A1: t = [x,y];
  (ex s being SortSymbol of S, v being Element of X.s st t.{} = [v,s])
  or t.{} in [:the carrier' of S,{the carrier of S}:]
  by MSATERM:2;
  then (ex s being SortSymbol of S, v being Element of X.s st t.{} = [v,s])
  or ex a,b being object st a in the carrier' of S &
  b in {the carrier of S} & t.{} = [a,b] by ZFMISC_1:def 2;
  then {{}} <> {{}, t.{}} by ZFMISC_1:5;
  then
A2: [{}, t.{}] <> {x} by ZFMISC_1:5;
  {} in dom t by TREES_1:22;
  then [{}, t.{}] in t by FUNCT_1:def 2;
  then
A3: [{}, t.{}] = {x,y} by A1,A2,TARSKI:def 2;
  x = y by A1,Th3;
  hence thesis by A2,A3,ENUMSET1:29;
end;
