let G be _Graph; :: thesis: for A, B being non empty Subset of (the_Vertices_of G) st B c= A holds

for H1 being inducedSubgraph of G,A

for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B

let A, B be non empty Subset of (the_Vertices_of G); :: thesis: ( B c= A implies for H1 being inducedSubgraph of G,A

for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B )

assume A1: B c= A ; :: thesis: for H1 being inducedSubgraph of G,A

for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B

let H1 be inducedSubgraph of G,A; :: thesis: for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B

let H2 be inducedSubgraph of H1,B; :: thesis: H2 is inducedSubgraph of G,B

A2: the_Vertices_of H1 = A by GLIB_000:def 37;

then A3: the_Vertices_of H2 = B by A1, GLIB_000:def 37;

then the_Edges_of H2 c= G .edgesBetween B by A1, A2, GLIB_000:def 37;

then for x being object holds

( x in the_Edges_of H2 iff x in G .edgesBetween B ) by A4;

then A10: the_Edges_of H2 = G .edgesBetween B by TARSKI:2;

then the_Edges_of H2 c= the_Edges_of G by XBOOLE_1:1;

then H2 is Subgraph of G by A3, A11, GLIB_000:def 32;

hence H2 is inducedSubgraph of G,B by A3, A10, GLIB_000:def 37; :: thesis: verum

for H1 being inducedSubgraph of G,A

for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B

let A, B be non empty Subset of (the_Vertices_of G); :: thesis: ( B c= A implies for H1 being inducedSubgraph of G,A

for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B )

assume A1: B c= A ; :: thesis: for H1 being inducedSubgraph of G,A

for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B

let H1 be inducedSubgraph of G,A; :: thesis: for H2 being inducedSubgraph of H1,B holds H2 is inducedSubgraph of G,B

let H2 be inducedSubgraph of H1,B; :: thesis: H2 is inducedSubgraph of G,B

A2: the_Vertices_of H1 = A by GLIB_000:def 37;

then A3: the_Vertices_of H2 = B by A1, GLIB_000:def 37;

A4: now :: thesis: for e being set st e in G .edgesBetween B holds

e in the_Edges_of H2

H1 .edgesBetween B c= G .edgesBetween B
by GLIB_000:76;e in the_Edges_of H2

let e be set ; :: thesis: ( e in G .edgesBetween B implies e in the_Edges_of H2 )

assume A5: e in G .edgesBetween B ; :: thesis: e in the_Edges_of H2

A6: (the_Target_of G) . e in B by A5, GLIB_000:31;

A7: (the_Source_of G) . e in B by A5, GLIB_000:31;

then e in G .edgesBetween A by A1, A5, A6, GLIB_000:31;

then A8: e in the_Edges_of H1 by GLIB_000:def 37;

then A9: (the_Target_of H1) . e = (the_Target_of G) . e by GLIB_000:def 32;

(the_Source_of H1) . e = (the_Source_of G) . e by A8, GLIB_000:def 32;

then e in H1 .edgesBetween B by A7, A6, A8, A9, GLIB_000:31;

hence e in the_Edges_of H2 by A1, A2, GLIB_000:def 37; :: thesis: verum

end;assume A5: e in G .edgesBetween B ; :: thesis: e in the_Edges_of H2

A6: (the_Target_of G) . e in B by A5, GLIB_000:31;

A7: (the_Source_of G) . e in B by A5, GLIB_000:31;

then e in G .edgesBetween A by A1, A5, A6, GLIB_000:31;

then A8: e in the_Edges_of H1 by GLIB_000:def 37;

then A9: (the_Target_of H1) . e = (the_Target_of G) . e by GLIB_000:def 32;

(the_Source_of H1) . e = (the_Source_of G) . e by A8, GLIB_000:def 32;

then e in H1 .edgesBetween B by A7, A6, A8, A9, GLIB_000:31;

hence e in the_Edges_of H2 by A1, A2, GLIB_000:def 37; :: thesis: verum

then the_Edges_of H2 c= G .edgesBetween B by A1, A2, GLIB_000:def 37;

then for x being object holds

( x in the_Edges_of H2 iff x in G .edgesBetween B ) by A4;

then A10: the_Edges_of H2 = G .edgesBetween B by TARSKI:2;

A11: now :: thesis: for e being set st e in the_Edges_of H2 holds

( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e )

the_Edges_of H2 c= the_Edges_of H1
;( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e )

let e be set ; :: thesis: ( e in the_Edges_of H2 implies ( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e ) )

assume A12: e in the_Edges_of H2 ; :: thesis: ( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e )

A13: (the_Target_of H2) . e = (the_Target_of H1) . e by A12, GLIB_000:def 32;

(the_Source_of H2) . e = (the_Source_of H1) . e by A12, GLIB_000:def 32;

hence ( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e ) by A12, A13, GLIB_000:def 32; :: thesis: verum

end;assume A12: e in the_Edges_of H2 ; :: thesis: ( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e )

A13: (the_Target_of H2) . e = (the_Target_of H1) . e by A12, GLIB_000:def 32;

(the_Source_of H2) . e = (the_Source_of H1) . e by A12, GLIB_000:def 32;

hence ( (the_Source_of H2) . e = (the_Source_of G) . e & (the_Target_of H2) . e = (the_Target_of G) . e ) by A12, A13, GLIB_000:def 32; :: thesis: verum

then the_Edges_of H2 c= the_Edges_of G by XBOOLE_1:1;

then H2 is Subgraph of G by A3, A11, GLIB_000:def 32;

hence H2 is inducedSubgraph of G,B by A3, A10, GLIB_000:def 37; :: thesis: verum