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Let R be a regular semi-local domain containing a field such that all theresidue fields are infinite. Let K be the fraction field of R. Let q be aquadratic space over R on a free rank n R-module P such that the projectivequadric q=0 is smooth over R. It is proved that if the quadratic space q isisotropic over K, then there is a unimodular vector v in the free rank nR-module P such that q(v)=0.  If characteristic of R is 2, then in the case of even n our assumption on qis equivalent to the one that q is a non-singular space in the sense of\cite{Kn} and in the case of odd n > 2 our assumption on q is equivalent to theone that q is a semi-regular in the sense of \cite{Kn}.

Rationally trivial quadratic spaces are locally trivial:III