Some textbooks such as Lawrence call the relationship between c 0 and c 1 stated in theorem 2-1 as "homotopic" and the function H : [0, 1] × [0, 1] → U as "homotopy between c 0 and c 1 ". However, "homotopic" or "homotopy" in above-mentioned sense are different (stronger than) typical definitions of "homotopic" or "homotopy"; the latter omit condition [TLH3]. So from now on we refer to homotopy (homotope) in the sense of theorem 2-1 as a tubular homotopy (resp. tubular-homotopic).