theorem Th144:
  A c= B iff a++A c= a++B
proof
  thus A c= B implies a++A c= a++B by Th47;
  assume
A1: a++A c= a++B;
  let z;
  assume z in A;
  then a+z in a++A by Th141;
  then ex c st a+z = a+c & c in B by A1,Th143;
  hence thesis;
end;
