reserve G for non empty addLoopStr;
reserve x for Element of G;
reserve M for non empty MidStr;
reserve p,q,r for Point of M;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve S for non empty set;
reserve a,b,b9,c,c9,d for Element of S;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable Abelian non
  empty addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;
reserve M for MidSp;
reserve p,q,r,s for Point of M;
reserve G for midpoint_operator add-associative right_zeroed
  right_complementable Abelian non empty addLoopStr;
reserve x,y for Element of G;
reserve x,y for Element of vectgroup(M);
reserve w for Function of [:S,S:],the carrier of G;
reserve a,b,c for Point of MidStr(#S,@(w)#);
reserve M for non empty MidStr;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve a,b,b1,b2,c for Point of M;
reserve M for MidSp;
reserve W for ATLAS of M;
reserve a,b,b1,b2,c,d for Point of M;
reserve x for Vector of W;

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;
