reserve x,y,z for set;

theorem
  for S being non void Signature for X being non-empty ManySortedSet of
  the carrier of S holds x is Element of FreeMSA X iff x is Term of S, X
proof
  let S be non void Signature;
  let X be non-empty ManySortedSet of the carrier of S;
A1: S-Terms X = TS DTConMSA X by MSATERM:def 1
    .= union rng FreeSort X by MSAFREE:11
    .= Union FreeSort X by CARD_3:def 4;
  FreeMSA X = MSAlgebra(# FreeSort X, FreeOper X #) by MSAFREE:def 14;
  hence thesis by A1;
end;
