reserve p,q,r for FinSequence,
  x,y for object;

theorem
  for R being Relation holds
  R is locally-confluent iff R commutes-weakly_with R
proof
  let R be Relation;
  hereby
    assume
A1: R is locally-confluent;
    thus R commutes-weakly_with R
    proof
      let a,b,c be object;
      assume [a,b] in R & [a,c] in R;
      then b,c are_convergent_wrt R by A1;
      hence thesis;
    end;
  end;
  assume
A2: for a,b,c being object st [a,b] in R & [a,c] in R
     ex d being object st R reduces b,d & R reduces c,d;
  let a,b,c be object;
  assume [a,b] in R & [a,c] in R;
  hence ex d being object st R reduces b,d & R reduces c,d by A2;
end;
