let F be PGraphMapping of G1,G2; :: thesis: ( F is semi-Dcontinuous implies ( F is directed & F is semi-continuous ) )
assume A1: F is semi-Dcontinuous ; :: thesis: ( F is directed & F is semi-continuous )
hence F is directed by Th16; :: thesis: F is semi-continuous
let e, v, w be object ; :: according to GLIB_010:def 15 :: thesis: ( e in dom (F _E) & v in dom (F _V) & w in dom (F _V) & (F _E) . e Joins (F _V) . v,(F _V) . w,G2 implies e Joins v,w,G1 )
assume A2: ( e in dom (F _E) & v in dom (F _V) & w in dom (F _V) ) ; :: thesis: ( not (F _E) . e Joins (F _V) . v,(F _V) . w,G2 or e Joins v,w,G1 )
assume (F _E) . e Joins (F _V) . v,(F _V) . w,G2 ; :: thesis: e Joins v,w,G1
per cases then ( (F _E) . e DJoins (F _V) . v,(F _V) . w,G2 or (F _E) . e DJoins (F _V) . w,(F _V) . v,G2 ) by GLIB_000:16;
suppose (F _E) . e DJoins (F _V) . v,(F _V) . w,G2 ; :: thesis: e Joins v,w,G1
then e DJoins v,w,G1 by A1, A2;
hence e Joins v,w,G1 by GLIB_000:16; :: thesis: verum
end;
suppose (F _E) . e DJoins (F _V) . w,(F _V) . v,G2 ; :: thesis: e Joins v,w,G1
then e DJoins w,v,G1 by A1, A2;
hence e Joins v,w,G1 by GLIB_000:16; :: thesis: verum
end;
end;