let G3, G4 be _Graph; :: thesis: for v1, v2 being object

for G1 being addVertex of G3,v1

for G2 being addVertex of G4,v2

for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 object ; :: thesis: for G1 being addVertex of G3,v1

for G2 being addVertex of G4,v2

for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 addVertex of G3,v1; :: thesis: for G2 being addVertex of G4,v2

for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 addVertex of G4,v2; :: thesis: for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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: ( not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 implies ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 ( not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 ) ; :: thesis: ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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) +* (v1 .--> v2)),(F0 _E)] 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 Th148;

take F ; :: thesis: ( F = [((F0 _V) +* (v1 .--> v2)),(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) +* (v1 .--> v2)),(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

for G1 being addVertex of G3,v1

for G2 being addVertex of G4,v2

for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 object ; :: thesis: for G1 being addVertex of G3,v1

for G2 being addVertex of G4,v2

for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 addVertex of G3,v1; :: thesis: for G2 being addVertex of G4,v2

for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 addVertex of G4,v2; :: thesis: for F0 being PGraphMapping of G3,G4 st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 holds

ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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: ( not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 implies ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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 ( not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 ) ; :: thesis: ex F being PGraphMapping of G1,G2 st

( F = [((F0 _V) +* (v1 .--> v2)),(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) +* (v1 .--> v2)),(F0 _E)] 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 Th148;

take F ; :: thesis: ( F = [((F0 _V) +* (v1 .--> v2)),(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) +* (v1 .--> v2)),(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