reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve u for UnOp of A,
  o,o9 for BinOp of A,
  a,b,c,e,e1,e2 for Element of A;

theorem Th11:
  o9 is_distributive_wrt o iff for a,b,c holds o9.(a,o.(b,c)) = o.
  (o9.(a,b),o9.(a,c)) & o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c))
proof
  thus o9 is_distributive_wrt o implies for a,b,c holds o9.(a,o.(b,c)) = o.(o9
  .(a,b),o9.(a,c)) & o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c))
  proof
    assume o9 is_left_distributive_wrt o & o9 is_right_distributive_wrt o;
    hence thesis;
  end;
  assume for a,b,c holds o9.(a,o.(b,c)) = o.(o9.(a,b),o9.(a,c)) & o9.(o.(a,b)
  ,c) = o.(o9.(a,c),o9.(b,c));
  hence (for a,b,c holds o9.(a,o.(b,c)) = o.(o9.(a,b),o9.(a,c))) & for a,b,c
  holds o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c));
end;
