Hall's Theorem(marriage theorem) IC{1,2,,m,JieI S≥|Il >S1;S2,...,Sm have a SDR case.2:there is a critical family in S1,S2,...,S say Sm-k+1U...USml=k k<m due to I.H.Sm-k+1,...,Sm have a SDR X=x1,...,x Si'-SX i=1,2,,m-k I{1,2,.,m-k}, Uim-k+1S:UUiEI S:k+ →UerS≥IS1, S2,...,Sm have a SDR Hall’s Theorem (marriage theorem) ⇥I ￾ {1, 2,...,m}, ￾ ￾ ⇥ i￾I Si ￾ ￾ ￾ |I|. case.2: there is a critical family in S1, S2, ..., Sm say k < m due to I.H. Sm-k+1,..., Sm have a SDR X={x1, ..., xk} |Sm￾k+1 ⇥ ··· ⇥ Sm| = k ⇤I ⇥ {1, 2,...,m ￾ k}, ￾ ￾ ⇥m i=m￾k+1 Si ￾ ⇥ i⇥I Si ￾ ￾ ￾ k + |I| ￾ |I| ￾ ￾ ⇥ i⇥I S￾ i ￾ ￾ Si ’ = Si\X i = 1, 2, ..., m-k