
theorem Th157:
  for G3, G4 being _Graph, v3 being Vertex of G3, v4 being Vertex of G4
  for e1,e2,v1,v2 being object
  for G1 being addAdjVertex of G3,v3,e1,v1
  for G2 being addAdjVertex of G4,v4,e2,v2
  for F0 being PGraphMapping of G3, G4
  st not e1 in the_Edges_of G3 & not e2 in the_Edges_of G4 &
    not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 &
    v3 in dom F0_V & F0_V.v3 = v4
  ex F being PGraphMapping of G1, G2 st
    F = [F0_V +* (v1 .--> v2), F0_E +* (e1 .--> e2)] &
    (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)
proof
  let G3, G4 be _Graph, v3 be Vertex of G3, v4 be Vertex of G4;
  let e1,e2,v1,v2 be object;
  let G1 be addAdjVertex of G3,v3,e1,v1, G2 be addAdjVertex of G4,v4,e2,v2;
  let F0 be PGraphMapping of G3, G4;
  assume that A1: not e1 in the_Edges_of G3 & not e2 in the_Edges_of G4 &
    not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 and
    A2: v3 in dom F0_V & F0_V.v3 = v4;
  consider G5 being addVertex of G3,v1 such that
    A3: G1 is addEdge of G5,v3,e1,v1 by A1, GLIB_006:125;
  consider G6 being addVertex of G4,v2 such that
    A4: G2 is addEdge of G6,v4,e2,v2 by A1, GLIB_006:125;
  consider F1 being PGraphMapping of G5, G6 such that
    A5: F1 = [F0_V +* (v1 .--> v2), F0_E] and
    A6: F0 is total implies F1 is total and
    A7: F0 is onto implies F1 is onto and
    A8: F0 is one-to-one implies F1 is one-to-one and
    A9: F0 is directed implies F1 is directed and
    F0 is semi-continuous implies F1 is semi-continuous and
    F0 is continuous implies F1 is continuous and
    F0 is semi-Dcontinuous implies F1 is semi-Dcontinuous and
    F0 is Dcontinuous implies F1 is Dcontinuous by A1, Th148;
  A10: v1 in dom F1_V & v3 in dom F1_V & F1_V.v1 = v2 & F1_V.v3 = v4
  proof
    v1 in {v1} by TARSKI:def 1;
    then v1 in dom({v1} --> v2);
    then A11: v1 in dom(v1 .--> v2) by FUNCOP_1:def 9;
    hence v1 in dom F1_V by A5, FUNCT_4:10, TARSKI:def 3;
    thus v3 in dom F1_V by A2, A5, FUNCT_4:10, TARSKI:def 3;
    thus F1_V.v1 = (v1 .--> v2).v1 by A5, A11, FUNCT_4:13
      .= v2 by FUNCOP_1:72;
    v3 <> v1 by A1;
    then not v3 in {v1} by TARSKI:def 1;
    then not v3 in dom(v1 .--> v2);
    hence F1_V.v3 = v4 by A2, A5, FUNCT_4:11;
  end;
  A12: not e1 in the_Edges_of G5 & not e2 in the_Edges_of G6
    by A1, GLIB_006:def 10;
  v2 in rng F1_V & v4 in rng F1_V by A10, FUNCT_1:3;
  then A13: v2 is Vertex of G6 & v4 is Vertex of G6;
  then consider F2 being PGraphMapping of G1, G2 such that
    A14: F2 = [F1_V, F1_E +* (e1 .--> e2)] and
    A15: F1 is total implies F2 is total and
    A16: F1 is onto implies F2 is onto and
    A17: F1 is one-to-one implies F2 is one-to-one
    by A3, A4, A10, A12, Th152;
  take F2;
  thus F2 = [F0_V +* (v1 .--> v2), F0_E +* (e1 .--> e2)] by A5, A14;
  thus F0 is total implies F2 is total by A6, A15;
  thus F0 is onto implies F2 is onto by A7, A16;
  thus F0 is one-to-one implies F2 is one-to-one by A8, A17;
  consider F3 being PGraphMapping of G1, G2 such that
    A18: F3 = [F1_V, F1_E +* (e1 .--> e2)] and
    A19: F1 is directed implies F3 is directed and
    F1 is Disomorphism implies F3 is Disomorphism
    by A3, A4, A10, A12, A13, Th154;
  thus F0 is directed implies F2 is directed by A9, A14, A18, A19;
end;
