let G1, G2 be _Graph; :: thesis: for F0 being PGraphMapping of G1,G2 st F0 _E is one-to-one holds
ex E being Subset of (the_Edges_of G2) st
for G3 being reverseEdgeDirections of G2,E ex F being PGraphMapping of G1,G3 st
( F = F0 & F is directed & ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) )
let F0 be PGraphMapping of G1,G2; :: thesis: ( F0 _E is one-to-one implies ex E being Subset of (the_Edges_of G2) st
for G3 being reverseEdgeDirections of G2,E ex F being PGraphMapping of G1,G3 st
( F = F0 & F is directed & ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) ) )
assume F0 _E is one-to-one ; :: thesis: ex E being Subset of (the_Edges_of G2) st
for G3 being reverseEdgeDirections of G2,E ex F being PGraphMapping of G1,G3 st
( F = F0 & F is directed & ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) )
then consider E being Subset of (the_Edges_of G2) such that
A1: for G3 being reverseEdgeDirections of G2,E ex F being PGraphMapping of G1,G3 st
( F = F0 & F is directed & ( not F0 is empty implies not F is empty ) & ( F0 is total implies F is total ) & ( F0 is one-to-one implies F is one-to-one ) & ( F0 is onto implies F is onto ) & ( F0 is semi-continuous implies F is semi-continuous ) & ( F0 is continuous implies F is continuous ) ) by Th92;
take E ; :: thesis: for G3 being reverseEdgeDirections of G2,E ex F being PGraphMapping of G1,G3 st
( F = F0 & F is directed & ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) )
let G3 be reverseEdgeDirections of G2,E; :: thesis: ex F being PGraphMapping of G1,G3 st
( F = F0 & F is directed & ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) )
consider F being PGraphMapping of G1,G3 such that
A2: ( F = F0 & F is directed ) and
( not F0 is empty implies not F is empty ) and
A3: ( F0 is total implies F is total ) and
A4: ( F0 is one-to-one implies F is one-to-one ) and
A5: ( F0 is onto implies F is onto ) and
( F0 is semi-continuous implies F is semi-continuous ) and
A6: ( F0 is continuous implies F is continuous ) by A1;
take F ; :: thesis: ( F = F0 & F is directed & ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) )
thus ( F = F0 & F is directed ) by A2; :: thesis: ( ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) )
thus ( ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) & ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) ) by A3, A4, A5, A6; :: thesis: verum