theorem
  (for a,x ex b st W.(a,b) = x) & (for a,b,c holds W.(a,b) = W.(a,c)
  implies b = c) & for a,b,c holds W.(a,b) + W.(b,c) = W.(a,c)
proof
  set w = the function of W;
  thus for a,x ex b st W.(a,b) = x
  proof
    set w = the function of W;
    let a,x;
    w is_atlas_of the carrier of M,(the algebra of W) by Def12;
    then consider b such that
A1: w.(a,b) = x;
    take b;
    thus thesis by A1;
  end;
  thus for a,b,c holds W.(a,b) = W.(a,c) implies b = c
  proof
    set w = the function of W;
A2: w is_atlas_of (the carrier of M),(the algebra of W) by Def12;
    let a,b,c;
    assume W.(a,b) = W.(a,c);
    hence thesis by A2;
  end;
  let a,b,c;
  w is_atlas_of the carrier of M,(the algebra of W) by Def12;
  hence thesis;
end;
