let G1, G2 be _Graph; :: thesis: ( <*G1,G2*> is edge-disjoint iff the_Edges_of G1 misses the_Edges_of G2 )
set F = <*G1,G2*>;
hereby :: thesis: ( the_Edges_of G1 misses the_Edges_of G2 implies <*G1,G2*> is edge-disjoint )
assume A1: <*G1,G2*> is edge-disjoint ; :: thesis: the_Edges_of G1 misses the_Edges_of G2
( 1 in {1,2} & 2 in {1,2} ) by TARSKI:def 2;
then reconsider x1 = 1, x2 = 2 as Element of dom <*G1,G2*> by FINSEQ_1:92;
the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2) by A1;
then the_Edges_of G1 misses the_Edges_of (<*G1,G2*> . x2) by FINSEQ_1:44;
hence the_Edges_of G1 misses the_Edges_of G2 by FINSEQ_1:44; :: thesis: verum
end;
assume A2: the_Edges_of G1 misses the_Edges_of G2 ; :: thesis: <*G1,G2*> is edge-disjoint
let x1, x2 be Element of dom <*G1,G2*>; :: according to GLIB_015:def 23 :: thesis: ( x1 <> x2 implies the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2) )
assume A3: x1 <> x2 ; :: thesis: the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2)
( x1 in dom <*G1,G2*> & x2 in dom <*G1,G2*> ) ;
then ( x1 in {1,2} & x2 in {1,2} ) by FINSEQ_1:92;
then ( ( x1 = 1 or x1 = 2 ) & ( x2 = 1 or x2 = 2 ) ) by TARSKI:def 2;
per cases then ( ( x1 = 1 & x2 = 2 ) or ( x1 = 2 & x2 = 1 ) ) by A3;
suppose ( x1 = 1 & x2 = 2 ) ; :: thesis: the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2)
then ( <*G1,G2*> . x1 = G1 & <*G1,G2*> . x2 = G2 ) by FINSEQ_1:44;
hence the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2) by A2; :: thesis: verum
end;
suppose ( x1 = 2 & x2 = 1 ) ; :: thesis: the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2)
then ( <*G1,G2*> . x2 = G1 & <*G1,G2*> . x1 = G2 ) by FINSEQ_1:44;
hence the_Edges_of (<*G1,G2*> . x1) misses the_Edges_of (<*G1,G2*> . x2) by A2; :: thesis: verum
end;
end;