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If ψ, σ, ρ are linear mappings out of a projective space (P,G) into a projective space (P', G') and ψ ≠ σ, then ρ is said to belong to the pencil <ψ,<ψ> of linear mappings spanned by ψ and σ if in the main ψ (x), σ (x), ρ (x) are collinear for all x ε P. We give some sufficient conditions for x ε P and ψ, σ, such that ρ (x) is uniquely determined by giving ψ, σ and ρ(z), z ε P.