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 Th51:
  for S being non empty set, A being non-empty ManySortedSet of S
  for R being ManySortedRelation of A for E being MSEquivalence_Relation-like
  ManySortedRelation of A st R c= E for s being Element of S for a,b being
  Element of A.s st a,b are_convertible_wrt R.s holds [a,b] in E.s
proof
  let S be non empty set, A be non-empty ManySortedSet of S;
  let R be ManySortedRelation of A;
  let E be MSEquivalence_Relation-like ManySortedRelation of A such that
A1: R c= E;
  let s be Element of S;
A2: E.s is Equivalence_Relation of A.s by MSUALG_4:def 2;
  R.s c= E.s by A1;
  hence thesis by A2,Th40;
end;
