theorem Th142:
  a++A = {a+c: c in A}
proof
  thus a++A c= {a+c: c in A}
  proof
    let e be object;
    assume e in a++A;
    then consider c1,c2 such that
A1: e = c1+c2 and
A2: c1 in {a} and
A3: c2 in A;
    c1 = a by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  let e be object;
  assume e in {a+c: c in A};
  then ex c st e = a+c & c in A;
  hence thesis by Th141;
end;
