theorem Th32:
  W.(a,a) = 0.W & (W.(a,b) = 0.W implies a = b) & W.(a,b) = -W.(b,
a) & (W.(a,b) = W.(c,d) implies W.(b,a) = W.(d,c)) & (for b,x ex a st W.(a,b) =
x) & (W.(b,a) = W.(c,a) implies b = c) & (a@b = c iff W.(a,c) = W.(c,b)) & (a@b
  = c@d iff W.(a,d) = W.(c,b)) & (W.(a,b) = x iff (a,x).W = b)
proof
  set w = the function of W, G = the algebra of W;
A1: w is_atlas_of (the carrier of M),G by Def12;
A2: G is midpoint_operator add-associative right_zeroed right_complementable
  Abelian by Def12;
  hence W.(a,a) = 0.W by A1,Th2;
  thus W.(a,b) = 0.W implies a = b by A2,A1,Th3;
  thus W.(a,b) = -W.(b,a) by A2,A1,Th4;
  thus W.(a,b) = W.(c,d) implies W.(b,a) = W.(d,c) by A2,A1,Th5;
  thus for b,x ex a st W.(a,b) = x
  proof
    let b,x;
    consider a such that
A3: w.(a,b) = x by A2,A1,Th6;
    take a;
    thus thesis by A3;
  end;
  thus W.(b,a) = W.(c,a) implies b = c by A2,A1,Th7;
A4: w is associating by Def12;
  hence a@b = c iff W.(a,c) = W.(c,b);
  thus a@b = c@d iff W.(a,d) = W.(c,b) by A2,A4,A1,Th13;
  thus thesis by A1,Def4;
end;
