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

theorem
  for R being Relation, a,b,c being object st R reduces b,a & b,c
  are_divergent_wrt R or a,b are_divergent_wrt R & R reduces b,c holds a,c
  are_divergent_wrt R
proof
  let R be Relation, a,b,c be object;
  assume
A1: R reduces b,a & b,c are_divergent_wrt R or a,b are_divergent_wrt R &
  R reduces b,c;
  per cases by A1;
  suppose
A2: R reduces b,a & b,c are_divergent_wrt R;
    then consider d being object such that
A3: R reduces d,b and
A4: R reduces d,c;
    R reduces d,a by A2,A3,Th16;
    hence thesis by A4;
  end;
  suppose
A5: a,b are_divergent_wrt R & R reduces b,c;
    then consider d being object such that
A6: R reduces d,a and
A7: R reduces d,b;
    R reduces d,c by A5,A7,Th16;
    hence thesis by A6;
  end;
end;
