On the fundamental conjecture of $GLC$ I.

G. Gentzen has founded in his well known paper The author has generalized in his former paper The proposition " Every provable sequence in GLC (resp. G¹LC ) is provable without cut " was called the funda- mental conjecture of GLC (resp. G¹LC ), and it was shown that from this conjecture would follow the consistency of the analysis (resp. of the theory of real numbers).