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;

theorem Th10:
  for G being add-associative Abelian non empty addLoopStr, x,y,
  z,t being Element of G holds (x+y)+(z+t) = (x+z)+(y+t)
proof
  let G be add-associative Abelian non empty addLoopStr, x,y,z,t be Element
  of G;
  thus (x+y)+(z+t) = x+(y+(z+t)) by RLVECT_1:def 3
    .= x+(z+(y+t)) by RLVECT_1:def 3
    .= (x+z)+(y+t) by RLVECT_1:def 3;
end;
