Extending a d-cohomology class to D-cocycle

[kakarotyjn]

Hi!The details is in the attached file.

I don't understand the sentence which is bolded,why if we extend the D-cochain to a D-cocycle,then we can get a global form which restricts to the d-cohomology class?

Any hints?Thank you!

 

[kakarotyjn]

Maybe I ignore to explain that $\pi:E\rightarrow M$ is a sphere bundle with structure group $Diff (S^n)$