Per the story the Belgians joined freely with the French - have a feeling they didnt want to be out there on a limb all by themselves - and remember that a lot of Belgians have a historic tie to France. And the areas they took over in the Netherlands and Germany pretty much were against formations that had taken serious losses already - i.e. if the Dutch and Germans had pre-war formations and air power there is a good chance the French and Belgians would have taken a very very bloody nose
And considering the size of the French and Belgian formations they had to have taken some good sized losses during the invasion of both areas - none of the French formations in those areas is anywhere near pre-war manpower or equipment
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