let P be non empty ProofSystem ; :: thesis: for B being Subset of P st P is consistent & not P is paraconsistent & B is consistent holds
ex B1 being Subset of P st
( B c= B1 & B1 is maximally-consistent )
let B be Subset of P; :: thesis: ( P is consistent & not P is paraconsistent & B is consistent implies ex B1 being Subset of P st
( B c= B1 & B1 is maximally-consistent ) )
assume A1: ( P is consistent & not P is paraconsistent ) ; :: thesis: ( not B is consistent or ex B1 being Subset of P st
( B c= B1 & B1 is maximally-consistent ) )
assume A2: B is consistent ; :: thesis: ex B1 being Subset of P st
( B c= B1 & B1 is maximally-consistent )
consider S being finite Subset of P such that
A3: S is inconsistent by A1;
A5: for B1 being Subset of P holds
( B1 is consistent iff B1 is S -omitting ) by A3;
then consider B1 being Subset of P such that
A10: B c= B1 and
A11: B1 is S -maximally-omitting by A2, Th61;
take B1 ; :: thesis: ( B c= B1 & B1 is maximally-consistent )
thus B c= B1 by A10; :: thesis: B1 is maximally-consistent
thus B1 is maximally-consistent by A5, A11; :: thesis: verum