reserve S for non empty non void ManySortedSign,
  A for MSAlgebra over S;
reserve A for non-empty MSAlgebra over S;
reserve S for non empty non void ManySortedSign,
  A for non-empty MSAlgebra over S,
  R for ManySortedRelation of the Sorts of A;

theorem
  for R being ManySortedRelation of A for s being SortSymbol of S, a,b
being Element of A,s holds [a,b] in (EqTh R).s iff a,b are_convertible_wrt (TRS
  R).s
proof
  let R be ManySortedRelation of A;
  let s be SortSymbol of S, a,b be Element of A,s;
  defpred Z[SortSymbol of S,set,set] means $2,$3 are_convertible_wrt (TRS R).
  $1;
  consider P being ManySortedRelation of the Sorts of A such that
A1: for s being SortSymbol of S for a,b being Element of A,s holds [a,b]
  in P.s iff Z[s,a,b] from MSRExistence;
  reconsider P as ManySortedRelation of A;
  reconsider P as EquationalTheory of A by A1,Th50;
  R c= P
  proof
    let i be object;
    assume i in the carrier of S;
    then reconsider s = i as SortSymbol of S;
    R.s c= P.s
    proof
      let x,y be object;
      assume
A2:   [x,y] in R.s;
      then reconsider a = x, b = y as Element of A,s by ZFMISC_1:87;
      R c= TRS R by Def13;
      then R.s c= (TRS R).s;
      then a,b are_convertible_wrt (TRS R).s by A2,REWRITE1:29;
      hence thesis by A1;
    end;
    hence thesis;
  end;
  then EqTh R c= P by Def15;
  then (EqTh R).s c= P.s;
  hence [a,b] in (EqTh R).s implies a,b are_convertible_wrt (TRS R).s by A1;
  R c= EqTh R by Def15;
  then TRS R c= EqTh R by Def13;
  hence thesis by Th51;
end;
