Conditional Probability Problem

Could you help me untangle this,please?
The book says......
It follows from the definition of conditional probability that
P(M/C int S)=P(C int M/S) / P(C/S)
Note,however,that C and S are independent so P(C/S)=P(C).
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I thought P(M/C int S)=P(C int S int M)/P(C int S) ,any ideas how to get the author's version?
Any help appreciated,thank you.

... P [ M / ( C∩S ) ]

= P [ M ∩ ( C ∩ S ) ] / P( C ∩ S )

= P [ ( C ∩ M ) ∩ S ] / P( C ∩ S ) ......... (1)
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Now,

P [ ( C ∩ M ) / S ] = P [ ( C ∩ M ) ∩ S ] / P( S )

so that

P [ ( C ∩ M ) ∩ S ] = P [ ( C ∩ M ) / S ] • P( S ) .............. (2)
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Also,

P ( C / S ) = P ( C ∩ S ) / P( S )

so that

P ( C ∩ S ) = P ( C / S ) • P( S ) ....................................... (3)
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Using ( 2 ) and (3) in (1),

P [ M / ( C ∩ S ) ] = P [ ( C ∩ M ) / S ] • P( S ) / P ( C / S ) • P( S )

. . . . . . . . . . . . . . = P [ ( C ∩ M ) / S ] / P ( C / S )
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HappyTo Help !
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