reserve x,y,z for set;

theorem Th34:
  for S being non void Signature for X being non-empty
  ManySortedSet of the carrier of S for t being Term of S,X for a being Element
  of rng t holds a = [a`1,a`2]
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;
  let a be Element of rng t;
  consider x being object such that
A1: x in dom t and
A2: a = t.x by FUNCT_1:def 3;
  reconsider x as Element of dom t by A1;
  a = (t|x).{} by A2,TREES_9:35;
  then (ex s being SortSymbol of S, v being Element of X.s st a = [v,s]) or a
  in [:the carrier' of S,{the carrier of S}:] by MSATERM:2;
  hence thesis by MCART_1:21;
end;
