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### Abstract

Our theories formalise various matrix properties that serve to
establish existence, uniqueness and characterisation of the solution
to affine systems of ordinary differential equations (ODEs). In
particular, we formalise the operator and maximum norm of matrices.
Then we use them to prove that square matrices form a Banach space,
and in this setting, we show an instance of Picard-LindelÃ¶fâ€™s
theorem for affine systems of ODEs. Finally, we use this formalisation
to verify three simple hybrid programs.