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

theorem Th42:
  for R being Relation, a,b,c being object st
  R reduces a,b & b,c are_convergent_wrt R or
  a,b are_convergent_wrt R & R reduces c,b holds a,c are_convergent_wrt R
proof
  let R be Relation, a,b,c be object;
  assume
A1: R reduces a,b & b,c are_convergent_wrt R or a,b are_convergent_wrt R
  & R reduces c,b;
  per cases by A1;
  suppose
A2: R reduces a,b & b,c are_convergent_wrt R;
    then consider d being object such that
A3: R reduces b,d and
A4: R reduces c,d;
    R reduces a,d by A2,A3,Th16;
    hence thesis by A4;
  end;
  suppose
A5: a,b are_convergent_wrt R & R reduces c,b;
    then consider d being object such that
A6: R reduces a,d and
A7: R reduces b,d;
    R reduces c,d by A5,A7,Th16;
    hence thesis by A6;
  end;
end;
