Hall's Theorem(marriage theorem) VI C {1,2,...,m},UierSi. S1,S2,...,Sm have a SDR case.I:there is no critical family in S1,S2,...,Sm I∈{1,2，，m}that |I<m,Jier S>lI take an arbitrary xE Sm as representative of Sm remove Si and x S;'=Sx}i=1,2,,m-1 I≤{1,2，，m-1},UerS≥I due to I.H.S1,...,Sm-1 have a SDR {1,...,m-1) 1,....m-1 and form a SDR for S1,S2,...,Smcase.1: there is no critical family in S1, S2, ..., Sm take an arbitrary x∈Sm as representative of Sm remove Sm and x Si ’ = Si\{x} i = 1, 2, ..., m-1 ⌅I ⇥ {1, 2,...,m ￾ 1}, ￾ ￾ ⇥ i⇥I S￾ i ￾ ￾ ⇤ |I| ⇥I ￾ {1, 2,...,m} that |I| < m, ￾ ￾ ⇥ i￾I Si ￾ ￾ > |I| due to I.H. S1, S 2,...,Sm have a SDR have a SDR Hall’s Theorem (marriage theorem) ⇥I ￾ {1, 2,...,m}, ￾ ￾ ⇥ i￾I Si ￾ ￾ ￾ |I|. S⇥ 1,...,S⇥ m￾1 have a SDR {x1,...,xm￾1} x1,...,xm￾1 and x form a SDR for S1, S2, ..., Sm