let G1, G2 be WGraph; :: thesis: for F being one-to-one PGraphMapping of G1,G2 holds

( F is weight-preserving iff F " is weight-preserving )

let F be one-to-one PGraphMapping of G1,G2; :: thesis: ( F is weight-preserving iff F " is weight-preserving )

thus ( F is weight-preserving implies F " is weight-preserving ) by Th2; :: thesis: ( F " is weight-preserving implies F is weight-preserving )

( F " is weight-preserving implies (F ") " is weight-preserving ) by Th2;

hence ( F " is weight-preserving implies F is weight-preserving ) by Th70; :: thesis: verum

( F is weight-preserving iff F " is weight-preserving )

let F be one-to-one PGraphMapping of G1,G2; :: thesis: ( F is weight-preserving iff F " is weight-preserving )

thus ( F is weight-preserving implies F " is weight-preserving ) by Th2; :: thesis: ( F " is weight-preserving implies F is weight-preserving )

( F " is weight-preserving implies (F ") " is weight-preserving ) by Th2;

hence ( F " is weight-preserving implies F is weight-preserving ) by Th70; :: thesis: verum