theorem Th133:
  f++F = {f+w: w in F}
proof
  thus f++F c= {f+w: w in F}
  proof
    let e be object;
    assume e in f++F;
    then consider w1,w2 such that
A1: e = w1+w2 and
A2: w1 in {f} and
A3: w2 in F;
    w1 = f by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  let e be object;
  assume e in {f+w: w in F};
  then ex w st e = f+w & w in F;
  hence thesis by Th132;
end;
