let G3, G4 be _Graph; :: thesis: for V1, V2 being set
for G1 being addVertices of G3,V1
for G2 being addVertices of G4,V2
for F0 being PGraphMapping of G3,G4
for f being one-to-one Function st dom f = V1 \ () & rng f = V2 \ () holds
ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )

let V1, V2 be set ; :: thesis: for G1 being addVertices of G3,V1
for G2 being addVertices of G4,V2
for F0 being PGraphMapping of G3,G4
for f being one-to-one Function st dom f = V1 \ () & rng f = V2 \ () holds
ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )

let G1 be addVertices of G3,V1; :: thesis: for G2 being addVertices of G4,V2
for F0 being PGraphMapping of G3,G4
for f being one-to-one Function st dom f = V1 \ () & rng f = V2 \ () holds
ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )

let G2 be addVertices of G4,V2; :: thesis: for F0 being PGraphMapping of G3,G4
for f being one-to-one Function st dom f = V1 \ () & rng f = V2 \ () holds
ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )

let F0 be PGraphMapping of G3,G4; :: thesis: for f being one-to-one Function st dom f = V1 \ () & rng f = V2 \ () holds
ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )

let f be one-to-one Function; :: thesis: ( dom f = V1 \ () & rng f = V2 \ () implies ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) ) )

assume ( dom f = V1 \ () & rng f = V2 \ () ) ; :: thesis: ex F being PGraphMapping of G1,G2 st
( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )

then consider F being PGraphMapping of G1,G2 such that
A1: F = [((F0 _V) +* f),(F0 _E)] and
( not F0 is empty implies not F is empty ) and
A2: ( F0 is total implies F is total ) and
A3: ( F0 is onto implies F is onto ) and
A4: ( F0 is one-to-one implies F is one-to-one ) and
A5: ( F0 is directed implies F is directed ) and
( F0 is semi-continuous implies F is semi-continuous ) and
A6: ( F0 is continuous implies F is continuous ) and
( F0 is semi-Dcontinuous implies F is semi-Dcontinuous ) and
( F0 is Dcontinuous implies F is Dcontinuous ) by Th144;
take F ; :: thesis: ( F = [((F0 _V) +* f),(F0 _E)] & ( 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )
thus F = [((F0 _V) +* f),(F0 _E)] by A1; :: 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 ) & ( F0 is Disomorphism implies F is Disomorphism ) )
thus ( F0 is weak_SG-embedding implies F is weak_SG-embedding ) by A2, A4; :: thesis: ( ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) & ( F0 is isomorphism implies F is isomorphism ) & ( F0 is Disomorphism implies F is Disomorphism ) )
thus ( F0 is strong_SG-embedding implies F is strong_SG-embedding ) by A2, A4, A6; :: thesis: ( ( F0 is isomorphism implies F is isomorphism ) & ( F0 is Disomorphism implies F is Disomorphism ) )
thus ( F0 is isomorphism implies F is isomorphism ) by A2, A3, A4; :: thesis: ( F0 is Disomorphism implies F is Disomorphism )
thus ( F0 is Disomorphism implies F is Disomorphism ) by A2, A3, A4, A5; :: thesis: verum