Solution lifting method for handling Meta-variables in the TH$\exists$OREM$\forall$ system

We approach the problem of finding witnessing terms in proofs by the method of meta-variables. We describe an efficient method for handling meta-variables in natural style proofs and its implementation in the TH$\exists$OREM$\forall$ system. The method is based on a special technique for finding meta-substitutions when the proof search is performed in an AND/OR deduction tree. The implementation does not depend on the search strategy and allows easy integration with various special provers as well as with special unification/solving engines. We demonstrate the use of this method in the context of a special forward/backward inference strategy for producing proofs in elementary analysis.