let G1 be non-Dmulti _Graph; :: thesis: for G2 being _Graph

for F being directed PGraphMapping of G1,G2 st F _V is one-to-one holds

F _E is one-to-one

let G2 be _Graph; :: thesis: for F being directed PGraphMapping of G1,G2 st F _V is one-to-one holds

F _E is one-to-one

let F be directed PGraphMapping of G1,G2; :: thesis: ( F _V is one-to-one implies F _E is one-to-one )

assume A1: F _V is one-to-one ; :: thesis: F _E is one-to-one

for F being directed PGraphMapping of G1,G2 st F _V is one-to-one holds

F _E is one-to-one

let G2 be _Graph; :: thesis: for F being directed PGraphMapping of G1,G2 st F _V is one-to-one holds

F _E is one-to-one

let F be directed PGraphMapping of G1,G2; :: thesis: ( F _V is one-to-one implies F _E is one-to-one )

assume A1: F _V is one-to-one ; :: thesis: F _E is one-to-one

now :: thesis: for e1, e2 being object st e1 in dom (F _E) & e2 in dom (F _E) & (F _E) . e1 = (F _E) . e2 holds

e1 = e2

hence
F _E is one-to-one
by FUNCT_1:def 4; :: thesis: verume1 = e2

let e1, e2 be object ; :: thesis: ( e1 in dom (F _E) & e2 in dom (F _E) & (F _E) . e1 = (F _E) . e2 implies e1 = e2 )

set v1 = (the_Source_of G1) . e1;

set w1 = (the_Target_of G1) . e1;

set v2 = (the_Source_of G1) . e2;

set w2 = (the_Target_of G1) . e2;

assume A2: ( e1 in dom (F _E) & e2 in dom (F _E) & (F _E) . e1 = (F _E) . e2 ) ; :: thesis: e1 = e2

then A3: ( (the_Source_of G1) . e1 in dom (F _V) & (the_Target_of G1) . e1 in dom (F _V) & (the_Source_of G1) . e2 in dom (F _V) & (the_Target_of G1) . e2 in dom (F _V) ) by Th5;

A4: ( e1 DJoins (the_Source_of G1) . e1,(the_Target_of G1) . e1,G1 & e2 DJoins (the_Source_of G1) . e2,(the_Target_of G1) . e2,G1 ) by A2, GLIB_000:def 14;

then ( (F _E) . e1 DJoins (F _V) . ((the_Source_of G1) . e1),(F _V) . ((the_Target_of G1) . e1),G2 & (F _E) . e2 DJoins (F _V) . ((the_Source_of G1) . e2),(F _V) . ((the_Target_of G1) . e2),G2 ) by A2, A3, Def14;

then ( (F _V) . ((the_Source_of G1) . e1) = (F _V) . ((the_Source_of G1) . e2) & (F _V) . ((the_Target_of G1) . e1) = (F _V) . ((the_Target_of G1) . e2) ) by A2, GLIB_009:6;

then ( (the_Source_of G1) . e1 = (the_Source_of G1) . e2 & (the_Target_of G1) . e1 = (the_Target_of G1) . e2 ) by A1, A3, FUNCT_1:def 4;

hence e1 = e2 by A4, GLIB_000:def 21; :: thesis: verum

end;set v1 = (the_Source_of G1) . e1;

set w1 = (the_Target_of G1) . e1;

set v2 = (the_Source_of G1) . e2;

set w2 = (the_Target_of G1) . e2;

assume A2: ( e1 in dom (F _E) & e2 in dom (F _E) & (F _E) . e1 = (F _E) . e2 ) ; :: thesis: e1 = e2

then A3: ( (the_Source_of G1) . e1 in dom (F _V) & (the_Target_of G1) . e1 in dom (F _V) & (the_Source_of G1) . e2 in dom (F _V) & (the_Target_of G1) . e2 in dom (F _V) ) by Th5;

A4: ( e1 DJoins (the_Source_of G1) . e1,(the_Target_of G1) . e1,G1 & e2 DJoins (the_Source_of G1) . e2,(the_Target_of G1) . e2,G1 ) by A2, GLIB_000:def 14;

then ( (F _E) . e1 DJoins (F _V) . ((the_Source_of G1) . e1),(F _V) . ((the_Target_of G1) . e1),G2 & (F _E) . e2 DJoins (F _V) . ((the_Source_of G1) . e2),(F _V) . ((the_Target_of G1) . e2),G2 ) by A2, A3, Def14;

then ( (F _V) . ((the_Source_of G1) . e1) = (F _V) . ((the_Source_of G1) . e2) & (F _V) . ((the_Target_of G1) . e1) = (F _V) . ((the_Target_of G1) . e2) ) by A2, GLIB_009:6;

then ( (the_Source_of G1) . e1 = (the_Source_of G1) . e2 & (the_Target_of G1) . e1 = (the_Target_of G1) . e2 ) by A1, A3, FUNCT_1:def 4;

hence e1 = e2 by A4, GLIB_000:def 21; :: thesis: verum