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

theorem
  for R,Q being Relation st R commutes_with Q holds R commutes-weakly_with Q
proof
  let R,Q be Relation;
  assume
A1: for a,b,c being object st R reduces a,b & Q reduces a,c
   ex d being object st Q reduces b,d & R reduces c,d;
  let a,b,c be object;
  assume [a,b] in R & [a,c] in Q;
  then R reduces a,b & Q reduces a,c by Th15;
  hence thesis by A1;
end;
