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

theorem
  for R being Relation holds R is confluent iff R[*] is subcommutative
proof
  let R be Relation;
  hereby
    assume
A1: R is confluent;
    thus R[*] is subcommutative
    proof
      let a,b,c be object;
      assume [a,b] in R[*] & [a,c] in R[*];
      then R reduces a,b & R reduces a,c by Th20;
      then b,c are_divergent_wrt R;
      then b,c are_convergent_wrt R by A1;
      then consider d being object such that
A2:   R reduces b,d & R reduces c,d;
      take d;
      thus thesis by A2,Th20;
    end;
  end;
  assume
A3: for a,b,c being object st [a,b] in R[*] & [a,c] in R[*] holds b,c
  are_convergent<=1_wrt R[*];
  let a,b be object;
  given c being object such that
A4: R reduces c,a and
A5: R reduces c,b;
A6: [c,b] in R[*] or c = b by A5,Th20;
  [c,a] in R[*] or c = a by A4,Th20;
  then a,b are_convergent<=1_wrt R[*] by A3,A6;
  then a,b are_convergent_wrt R[*] by Th44;
  then consider d being object such that
A7: R[*] reduces a,d & R[*] reduces b,d;
  take d;
  thus thesis by A7,Th21;
end;
