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 Th39:
  for A being set for R,E being Relation of A st for a,b being set
  st a in A & b in A holds [a,b] in E iff a,b are_convertible_wrt R holds E is
  total symmetric transitive
proof
  let A be set;
  let R,E be Relation of A;
  set X = A;
  assume
A1: for a,b being set st a in A & b in A holds [a,b] in E iff a,b
  are_convertible_wrt R;
  now
    let x be object;
    x,x are_convertible_wrt R by REWRITE1:26;
    hence x in X implies [x,x] in E by A1;
  end;
  then
A2: E is_reflexive_in X by RELAT_2:def 1;
  then
A3: field E = X by ORDERS_1:13;
  dom E = X by A2,ORDERS_1:13;
  hence E is total by PARTFUN1:def 2;
  now
    let x,y be object;
    assume that
A4: x in X and
A5: y in X;
    assume [x,y] in E;
    then x,y are_convertible_wrt R by A1,A4,A5;
    then y,x are_convertible_wrt R by REWRITE1:31;
    hence [y,x] in E by A1,A4,A5;
  end;
  then E is_symmetric_in X by RELAT_2:def 3;
  hence E is symmetric by A3,RELAT_2:def 11;
  now
    let x,y,z be object;
    assume that
A6: x in X and
A7: y in X and
A8: z in X;
    assume that
A9: [x,y] in E and
A10: [y,z] in E;
A11: y,z are_convertible_wrt R by A1,A7,A8,A10;
    x,y are_convertible_wrt R by A1,A6,A7,A9;
    then x,z are_convertible_wrt R by A11,REWRITE1:30;
    hence [x,z] in E by A1,A6,A8;
  end;
  then E is_transitive_in X by RELAT_2:def 8;
  hence thesis by A3,RELAT_2:def 16;
end;
