
theorem Th148:
  for G3, G4 being _Graph, v1, v2 being object
  for G1 being addVertex of G3,v1, G2 being addVertex of G4,v2
  for F0 being PGraphMapping of G3,G4
  st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4
  ex F being PGraphMapping of G1, G2 st F = [F0_V +* (v1 .--> v2), F0_E] &
    (F0 is total implies F is total) &
    (F0 is onto implies F is onto) &
    (F0 is one-to-one implies F is one-to-one) &
    (F0 is directed implies F is directed) &
    (F0 is semi-continuous implies F is semi-continuous) &
    (F0 is continuous implies F is continuous) &
    (F0 is semi-Dcontinuous implies F is semi-Dcontinuous) &
    (F0 is Dcontinuous implies F is Dcontinuous)
proof
  let G3, G4 be _Graph, v1, v2 be object;
  let G1 be addVertex of G3,v1, G2 be addVertex of G4,v2;
  let F0 be PGraphMapping of G3,G4;
  assume A1: not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4;
  set f = v1 .--> v2;
  A2: dom f = dom({v1} --> v2) by FUNCOP_1:def 9
    .= {v1} \ the_Vertices_of G3 by A1, ZFMISC_1:59;
  v1 is set & v2 is set by TARSKI:1;
  then rng f = {v2} by FUNCOP_1:88
    .= {v2} \ the_Vertices_of G4 by A1, ZFMISC_1:59;
  then consider F being PGraphMapping of G1, G2 such that
    A3: F = [F0_V +* f, F0_E] &
    (F0 is non empty  implies F is non empty) &
    (F0 is total implies F is total) &
    (F0 is onto implies F is onto) &
    (F0 is one-to-one implies F is one-to-one) &
    (F0 is directed implies F is directed) &
    (F0 is semi-continuous implies F is semi-continuous) &
    (F0 is continuous implies F is continuous) &
    (F0 is semi-Dcontinuous implies F is semi-Dcontinuous) &
    (F0 is Dcontinuous implies F is Dcontinuous) by A2, Th144;
  take F;
  thus thesis by A3;
end;
