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

let e1, e2 be object ; :: thesis: ( e1 in dom (F _E) & e2 in dom (F _E) & (F _E) . e1 = (F _E) . e2 implies b₁ = b₂ )
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: b₁ = b₂
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 Joins (the_Source_of G1) . e1,(the_Target_of G1) . e1,G1 & e2 Joins (the_Source_of G1) . e2,(the_Target_of G1) . e2,G1 ) by A2, GLIB_000:def 13;
then ( (F _E) . e1 Joins (F _V) . ((the_Source_of G1) . e1),(F _V) . ((the_Target_of G1) . e1),G2 & (F _E) . e2 Joins (F _V) . ((the_Source_of G1) . e2),(F _V) . ((the_Target_of G1) . e2),G2 ) by A2, A3, Th4;

per cases 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) ) or ( (F _V) . ((the_Source_of G1) . e1) = (F _V) . ((the_Target_of G1) . e2) & (F _V) . ((the_Target_of G1) . e1) = (F _V) . ((the_Source_of G1) . e2) ) ) by A2, GLIB_000:15;

suppose ( (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) ) ; :: thesis: b₁ = b₂

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 20; :: thesis: verum

end;

suppose ( (F _V) . ((the_Source_of G1) . e1) = (F _V) . ((the_Target_of G1) . e2) & (F _V) . ((the_Target_of G1) . e1) = (F _V) . ((the_Source_of G1) . e2) ) ; :: thesis: b₁ = b₂

then ( (the_Source_of G1) . e1 = (the_Target_of G1) . e2 & (the_Target_of G1) . e1 = (the_Source_of G1) . e2 ) by A1, A3, FUNCT_1:def 4;
then e2 Joins (the_Source_of G1) . e1,(the_Target_of G1) . e1,G1 by A4, GLIB_000:14;
hence e1 = e2 by A4, GLIB_000:def 20; :: thesis: verum

end;

end;

end;

hence F _E is one-to-one by FUNCT_1:def 4; :: thesis: verum